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MATHEMATICAL   WORKS, 
By  professor   EDWARD  A.    BOWSER. 

ACADEMIC  ALGEBRA.    With  Numerous  Examples. 

COLLEGE  ALGEBRA.    With  Numerous  Examples. 

PLANE  AND  SOLID  GEOMETRY.  With  Numerous 
Exercises. 

ELEMENTS  OF  PLANE  AND  SPHERICAL  TRIG- 
ONOMETRY.   With  Numerous  Examples. 

A  TREATISE  ON  PLANE  AND  SPHERICAL  TRIG- 
ONOMETRY, AND  ITS  Applications  to  Astronomy 
AND  Geodesy.    With  Numerous  Examples. 

AN  ELEMENTARY  TREATISE  ON  ANALYTIC 
GEOMETRY,  Embracing  Plane  Geometry  and 
AN  Introduction  to  Geometry  of  Three  Dimen- 
sions.   With  Numerous  Examples. 

AN  ELEMENTARY  TREATISE  ON  THE  DIFFER- 
ENTIAL AND  INTEGRAL  CALCULUS.  With 
Numerous  Examples. 

AN  ELEMENTARY  TREATISE  ON  ANALYTIC 
MECHANICS.    With  Numerous  Examples. 

AN  ELEMENTARY  TREATISE  ON  HYDROME- 
CHANICS.   With  Numerous  Examples. 

LOGARITHMIC  AND   TRIGONOMETRIC  TABLES. 

A  TREATISE  ON  ROOFS  AND  BRIDGES.  With 
Numerous  Exercises. 


AN 


Elementary  Treatise 


ON 


HYDROMECHANICS, 


With  Numerous  Examples 


BY 


EDWARD    A.    BOWSER,    LL.  D., 

PROFESSOR     OF     MATHEMATICS    AND     ENGINEERING     IN     RUTGERS    COLLEGE. 


SEVENTH  EDITION 


NEW  YORK:    _ 

D.  VAN   NOSTRAND   COMPANY, 

EIGHT   WARREN   STREET. 
1921 


Copyright,  1885, 
By   E.    A.    BOWSER. 


jrri         Mathematical 
^/-)  /         fciences 

/rz/ 

PREFACE. 


ryiHE  present  work  on  Hydromechanics  is  designed  as  a 
-*-  text-book  for  Scientific  Schools  and  Colleges,  and  is 
prepared  on  the  same  general  plan  as  the  author's  Analytic 
Mechanics,  which  it  is  intended  to  follow.  Like  the  Ana- 
lytic Mechanics,  it  involves  the  use  of  Analytic  Geometry 
and  the  Calculus,  though  a  geometric  proof  has  been  intro- 
duced wherever  it  seemed  preferable. 

The  book  is  divided  into  two  parts,  namely.  Hydrostatics 
and  Hydrokinetics.  The  former  is  subdivided  into  three, 
and  the  latter  into  four  chapters ;  and  at  the  ends  of  the 
chapters  a  large  number  of  examples  is  given,  with  a  view 
to  illustrate  every  part  of  the  siibject.  Many  of  these  ex- 
amples were  prepared  specially  for  this  work,  and  are  prac- 
tical questions  in  hydrauUcs,  etc.,  taken  from  every-day  life. 

In  writing  this  treatise,  the  aim  has  been  to  enunciate 
clearly  the  fundamental  principles  of  the  theory  of  Hydro- 
mechanics, to  explain  some  of  the  most  important  applica- 
tions of  these  principles,  and  to  render  more  general  the 
study  of  this  interesting  science,  by  presenting  as  simple  a 
view  of  its  principles  as  is  consistent  with  scientific  accu- 
racy. Throughout  the  work  a  careful  distinction  has  been 
made  between  those  propositions  which  are  necessarily  true, 
being  deduced  from  the  definitions  and  axioms  of  the  sub- 
ject, and  those  results  which  are  empirical. 

8G6095 


IV  PREFACE. 

In  an  elementary  work  of  this  kind  there  is  not  room  for 
much  that  is  new.  I  have  drawn  freely  upon  the  writings 
of  many  of  the  best  authors.  The  works  to  which  I  am 
principally  indebted,  and  which  are  here  named  for  con- 
venience of  reference  by  the  student,  are  those  of  Besant, 
Lamb,  Eankine,  Boucharlat,  Weisbach,  Cotterill,  Bland, 
Jamieson,  Fanning,  Pratt,  Kenwick,  Stanley,  Tate,  Descha- 
nel,  Bossut,  d'Aubuisson,  Poncelet,  Eytelwein,  Prony, 
Starrow,  Goodeve,  Galbraith,  Gregory,  Twisden,  Bartlett, 
Wood,  Smith,  Olmsted,  Morin,  Humphreys  and  Abbot, 
Fairbairn,  Colyer,  Barrow,  and  the  Encyclopaedia  Britan- 
nica. 

My  thanks  are  again  due  to  my  friend  and  former  pupil, 
Mr.  E.  W.  Prentiss,  of  the  Nautical  Almanac  Office,  and 
formerly  Fellow  in  Mathematics  at  the  Johns  Hopkins 
University,  for  reading  the  MS.  and  for  valuable  sugges- 
tions. 

E.  A.  B. 
Rutgers  CoLiiEOE, 

New  Brunswick,  N.  J„  April,  1885. 


TABLE   OF   CONTENTS 


PART    I. 

HYDROSTATICS. 


CHAPTEE    I. 


EQUILIBRIUM   AXD    PRESSURE   OF  FLUIDS. 

ABT.  PAGS 

1.  Definitions — Hydrostatics,  Hydrokinetics 1 

2.  Three  States  of  Matter 1 

3.  A  Perfect  Fluid 2 

4.  Direction  of  Pressure 3 

5.  Solidifying  a  Fluid 4 

6.  Measure  of  the  Pressure  of  Fluids 4 

7.  Pressure  the  Same  in  Every  Direction 5 

8.  Equal  Transmission  of  Fluid  Pressure 6 

9.  Equilibrium  of  Pressures 8 

10.  Pressure  of  a  Liquid  at  any  Depth 10 

11.  Free  Surface  of  a  Liquid  at  Rest 13 

12.  Common  Surface  of  Two  Fluids 15 

13.  Two  Fluids  in  a  Bent  Tube 16 

14.  Pressure  on  Planes 17 

15.  The  Wliole  Pressure 18 

16.  Centre  of  Pressure 21 

17.  Embankments 27 

18.  Embankment  when  the  Face  on  the  Water  Side  is  Vertical. . .  27 

19.  Embankment  when  the  Face  on  the  Water  Side  is  Slanting . .  29 

20.  Pressure  upon  Both  Sides  of  a  Surface 33 

21.  Rotating  Liquid 35 

22.  Pressure  at  any  Point  of  a  Rotating  Liquid 37 

23.  Strength  of  Pipes  and  Boilers 39 

Examples 42 


VI  CONTENTS. 

CHAPTER    II. 

EQUILIBRI[JM   OF   FLOATING    BODIES — SPECIFIC    GRAVITY. 

ABT.  PAGE 

34.  Upward  Pressure,  Buoyant  Effort 50 

25.  Conditions  of  Equilibrium  of  an  Immersed  Solid 52 

26.  Depth  of  Flotation 54 

27.  Stability  of  Equilibrium 57 

28.  Position  of  the  Metacentre ;  Measure  of  Stability 60 

29.  Specific  Gravity 66 

30.  The  Standard  Temperature 67 

31.  Methods  of  Finding  Specific  Gravity 69 

32.  Specific  Gravity  of  a  Solid  Broken  into  Fragments 72 

33.  Specific  Gravity  of  Air 73 

34.  Specific  Gravity  of  a  Mixture 73 

35.  Weights  of  the  Components  of  a  Mechanical  Mixture 75 

36.  The  Hydrostatic  Balance 76 

37.  The  Common  Hydrometer 77 

38.  Sikes's  Hydrometer 78 

39.  Nicholson's  Hydrometer 79 

Examples 80 

CHAPTER    III. 

EQUILIBRIUM  AND  PRESSURE   OF  GASES — ELASTIC   FLUIDS. 

40.  Elasticity  of  Gases 88 

41.  Pressure  of  the  Atmosphere 89 

43.  Weight  of  the  Air 90 

43.  The  Barometer 91 

44.  The  Mean  Barometric  Height 93 

45.  The  Water-Barometer 93 

46.  Manometers 93 

47.  The  Atmospheric  Pressure  on  a  Square  Inch 94 

48.  Boyle  and  Mariotte's  Law 95 

49.  Effect  of  Heat  on  Gases 98 

50.  Thermometers — Fahrenheit,  Centigrade,  Reaumur 99 

51.  Comparison  of  the  Scales  of  these  Thermometers 100 

53.  Expansion  of  Mercury 101 

53.  Dalton's  and  Gay-Lussac's  Law 101 

54.  Pressure,  Temperature,  and  Density 108 

55.  Absolute  Temperature 105 


CONTENTS.  Vn 

ABT.  PAOE 

56.  The  Pressure  of  a  Mixture  of  Gases 106 

57.  Mixture  of  Equal  Volumes  of  Gases 107 

58.  Mixture  of  Unequal  Volumes  of  Gases 108 

59.  Vapors,  Gases 108 

60.  Formation  of  Vapor,  Saturation 109 

61.  Volume  of  Atmospheric  Air  without  its  Vapor 110 

63.  Change  of  Volume  and  Temperature 110 

63.  Formation  of  Dew — the  Dew  Point 112 

64.  Pressure  of  Vapor  in  the  Air 112 

65.  Effect  of  Compression  or  Dilatation  on  Temperature 113 

66.  Expansion  of  Bodies — Maximum  Density  of  Water 113 

67.  Thermal  Capacity — Unit  of  Heat — Specific  Heat 115 

68.  Specific   Heat   at   a   Constant   Pressure,   and   at   a   Constant 

Volume 116 

69.  Sudden  Compression  of  a  Mass  of  Air 118 

70.  Mass  of  the  Earth's  Atmosphere 120 

71.  The  Height  of  the  Homogeneous  Atmosphere 120 

73.  Necessary  Limit  to  the  Height  of  the  Atmosphere 121 

73.  Decrease  of  Density  of  the  Atmosphere 122 

74.  Heights  Determined  by  the  Barometer 124 

Table  of  Specific  Gravities 130 

Examples 131 


PART    II. 

HYDROKINETICS. 


CHAPTER    I. 

MOTION  OF  LIQUIDS — EFFLUX — RESISTANCE  AND  WORK  OF 
LIQUIDS. 

75.  Velocity  of  a  Liquid  in  Pipes 136 

76.  Velocity  of  Efflux 137 

77.  The  Horizontal  Ranjre 140 

78.  Time  of  Discharge  when  the  Height  is  Constant 141 


Viii  CONTENTS. 

ABT.  PAGE 

79.  Time  of  Em]>tying  any  Vessel 142 

80.  Time  of  Emptying  a  Cylinder  into  a  Vacuum 144 

81.  Time  of  Emptying  a  Paraboloid 145 

82.  Cylindrical  Vessel  with  Two  Small  OriSces 145 

83.  Orifice  in  the  Side  of  a  Conical  Vessel 146 

84.  Velocity  of  Efflux  through  an  Orifice  in  the  Bottom 147 

85.  Rectangular  Orifice  in  the  Side  of  a  Vessel 149 

86.  Triangular  Orifice  in  the  Side  of  a  Vessel 151 

87.  Time  of  Emptying'  any  Vessel  through  a  Vertical  Orifice. . . .  156 

88.  Efflux  from  a  Vessel  in  Motion 158 

89.  Efflux  from  a  Rotating  Vessel 160 

90.  The  Clepsydra,  or  Water-Clock 161 

91.  The  Vena  Contracta 162 

92.  Coefficient  of  Contraction 163 

93.  Coefficient  of  Veh)city 164 

94.  Coefficient  of  Efflux 164 

95.  Efflux  through  Sliort  Tubes,  or  Ajutages 165 

96.  Coefficient  of  Resistance 167 

97.  Resistance  and  Pressure  of  Fluids 170 

98.  Work  and  Pressure  of  a  Stream  of  Water 172 

99.  Impact  against  any  Surface  of  Revolution 174 

100.  Oblique  Impact 178 

101.  Maximum  Work  done  by  the  Impulse 180 

Examples 181 


CHAPTEE    II. 

MOTION^   OF   WATER   IJT   PIPES   AND   OPEN   CHANNELS. 

102.  Resistance  of  Friction 185 

103.  Motion  of  Water  in  Pipes 186 

104.  Uniform  Pipe  connecting  Two  Reservoirs 188 

105.  Coefficient  of  Friction  for  Pipes 191 

106.  The  Quantity  Discharged  from  Pipes 194 

107.  The  Diameter  of  Pipes 197 

108.  Sudden  Enlargement  of  Section 199 

109.  Sudden  Contraction  of  Section 201 

110.  Elbows 204 

111.  Bends 206 

\\\a    Equivalent  Pipes 207 

1116.  Discharge  Diminishing  Uniformly 208 


CONTENTS.  IX 

ART.  PAGE 

112.  General  Formula  for  all  the  Resistances 209 

113.  Flow  of  Water  in  Rivers  and  Canals 211 

114  Different  Velocities  in  a  Cro5S-Section 212 

115.  Transverse  Section  of  the  Stream 215 

116.  Mean  Velocity 216 

117.  Ratio  of  Mean  to  Greatest  Surface  Velocity 216 

118.  Processes  for  Gauging  Streams 218 

119.  Most  Economical  Form  of  Transverse  Section 221 

120.  Trapezoidal  Section  of  Least  Resistance 222 

131 .  Uniform  Motion 224 

122.  Coefficients  of  Friction 226 

123.  Variable  Motion 228 

124.  Bottom  Velocity  at  which  Scour  Commences 232 

125.  Transporting  Power  of  Water 233 

126.  Back  Water 235 

127.  River  Bends 236 

Examples 237 

CHAPTER    III. 

MOTION   OF   ELASTIC    FLUIDS. 

128.  Work  of  the  Expansion  of  Air 242 

129.  Velocity  of  Efflux  of  Air  according  to  Mariotte's  Law 244 

130.  Efflux  of  Moving  Air 247 

131.  Coefficient  of  Efflux 249 

132.  The  Quantity  Discharged 250 

133.  Coefficient  of  Friction  of  Air 251 

134   Motion  of  Air  in  Long  Pipes 252 

135.  Law  of  the  Expansion  of  Steam ' 254 

136.  Work  of  Expansion  of  Steam 256 

137.  Work  of  Steam  at  Efflux 257 

138.  Work  of  Steam  in  the  Expansive  Engine 259 

Examples 260 

CHAPTER    rtT 

HYDROSTATIC   AND    HYDRAULIC   MACHINES. 

139.  Definitions 263 

140.  The  Hydrostatic  Bellows ; 268 


X  CONTENTS. 

AST.  PAGE 

141.  The  Siphon 264 

142.  The  Diving  Bell 266 

143.  The  Common  Pump  (Suction  Pump) 268 

144  Tension  of  the  Piston-Rod 270 

145.  Height  through  which  Water  Rises  in  One  Stroke 271 

146.  The  Lifting  Pump 273 

147.  The  Forcing  Pump 274 

148.  The  Fire  Engine 276 

149.  Bramah's  Press ., 276 

150.  Hawksbee's  Air-Pump 277 

151.  Smeaton's  Air-Pump 279 

152.  The  Hydraulic  Ram 280 

153.  Work  of  Water  Wheels 282 

154.  Work  of  Overshot  Wheels 283 

155.  Work  of  Breast  Wheels 284 

156.  Work  of  Undershot  Wheels 285 

157.  Work  of  the  Poncelet  Water  Wheel 286 

158.  The  Reaction  Wheel ;  Barker's  Mill 288 

159.  The  Centrifugal  Pump 290 

160.  Turbines 292 

Examples 294 


HYDROMECHANICS. 


PART    I. 

HYDRQSTATICS, 


CHAPTER    I. 

EQUILIBRIUM    AND    PRESSURE    OF    FLUIDS. 

1.  Definitions.— Hydromechanics  is  the  science  wTiich 
treats  of  the  equilibrium  and  motion  of  fluids.  It  is  accord- 
ingly divided  into  two  parts,  Hydrostatics  and  Hydrokinetics. 

Hydrostatics  treats  of  the  equilibrium  of  fluids. 

Hydrokinetics  treats  of  the  motion  of  fluids. 

The  object  of  the  science  of  Hydrostatics  is  to  determine 
the  equilibrium  and  pressure  of  fluids,  the  nature  of  the 
action  which  fluids  exert  upon  one  another  and  upon  bodies 
with  which  they  are  in  contact,  and  the  weight  and  pressure 
of  solids  immersed  in  them,  and  to  explain  and  classify, 
under  general  laws,  the  different  phenomena  to  which  they 
give  rise. 

2.  Three  States  of  Matter Bodies  exist  in   three 

different  states,  depending  upon  the  manner  in  which  their 
particles  are  held  together.  They  are  either  solid  ov  fluid  ; 
and  the  latter  are  either  liquid  or  gaseous. 

Solid  bodies  are  those  whose  particles  are  held  together  so 
firmly  that  a  certain  force  is  necessary  to  change  their  forms 


2  A   PERFECT  FLUID. 

or  to  produce  a  separation  of  their  particles.  K  a  solid  be 
reduced  to  the  finest  powder,  still  each  grain  of  the  powder 
is  a  solid  body,  and  it-s  particles  are  held  together  in  a  de- 
terminate shape. 

Fluids  are  bodies,  the  position  of  whose  particles  in  ref- 
erence to  one  another  is  changed  by  the  smallest  force. 
The  distinguishing  property  of  a  fluid  is  the  perfect  facil- 
ity with  which  its  particles  move  among  one  another,  and 
as  a  consequence  its  readiness  to  change  its  form  under  the 
influence  of  the  slightest  effort. 

Fluids  are  of  two  kinds,  liquids  and  gases.  In  a  liquid 
there  is  a  perceptible  cohesion  among  its  particles ;  but  in 
a  gas  the  particles  mutually  rej^el  one  another.  Every  solid 
body  possesses  a  peculiar  form  of  its  own,  and  a  definite 
volume;  liquids  have  only  a  definite  volume,  but  no  pecu- 
liar form  ;  and  gases  have  neither  one  nor  the  other.  If  a 
liquid,  such  as  water,  be  poured  into  a  tumbler,  it  will  lie 
at  the  bottom,  and  will  be  separated  by  a  distinct  surface 
from  the  air  above  it;  but  if  ever  so  small  a  quantity  oi  gas 
be  introduced  into  an  empty  and  closed  vessel,  it  will  im- 
mediately expand  so  as  to  fill  the  whole  vessel,  and  will 
exert  some  amount  of  pressure  upon  the  interior  surface. 

3.  A  Perfect  Fluid.— Fluids  differ  from  each  other  in 
the  degree  of  cohesion  of  their  particles,  and  the  facility 
with  which  they  will  yield  to  the  action  of  a  force.  Many 
bodies  which  are  met  with  in  nature,  such  as  water,  mer- 
cury, air,  etc.,  possess  the  properties  of  fluids  in  an  eminent 
degree,  while  others,  such  as  oil,  tallow,  the  sirups,  etc., 
possess  a  less  degree  of  fluidity.  The  former  are  called 
perfect  fluids,  and  the  latter  viscous  or  imperfect  fluids. 
In  this  work,  only  perfect  fluids  will  be  considered. 

A  perfect  fluid  is  an  aggregation  of  paHicles  which 
yield  at  once  to  the  slightest  effort  made  to  separate 
them  from,  one  another. 


DIRECTION  OF  PRESSURE.  3 

Fluids  are  divided  into  two  classes,  incompressible  and 
compressible.  The  former  are  sometimes  called  inelastic 
and  the  latter  elastic  fluids. 

Incompressible  fluids  are  those  which  retain  the  same 
volume  under  a  variable  pressure.  Compressible  fluids  are 
those  in  which  the  volume  is  diminished  as  the  pressure 
upon  it  is  increased,  and  increased  as  the  pressure  upon  it 
is  diminished. 

The  term  incompressible  cannot  strictly  be  applied  to 
any  body  in  nature,  all  being  more  or  less  compressible. 
But  on  account  of  the  enormous  power  required  to  change, 
in  any  sensible  degree,  the  volumes  of  liquids,  they  are 
treated  in  most  of  the  researches  in  hydrostatics  as  incom- 
pressible or  inelastic  fluids.  It  was  shown  by  Canton,  in 
1761,  that  water  under  a  pressure  of  one  atmosphere,  i.  e., 
of  about  one  ton  on  each  square  foot  of  surface,  undergoes 
a  diminution  of  forty-four  milUonths  of  its  total  volume.* 
All  ?1(7m/(/s  are  therefore  regarded  as  incompressible.  Water, 
mercury,  wine,  etc.,  are  generally  ranged  under  this  class. 
The  (/ases  are  highly  compressible,  such  as  air  and  the  dif- 
ferent vapors. 

4.  The  Direction  of  the  Pressure  of  a  Fluid  on 
a  Surface. — If  an  indefinitely  thin  plate  be  made  to  di- 
vide a  fluid  in  any  direction,  no  resistance  will  be  oflfered 
to  the  motion  of  the  plate  in  the  direction  of  its  plane,  i.  e., 
there  will  be  no  tangential  resistance  of  the  nature  of  fric- 
tion, such,  for  instance,  as  would  be  exert,ed  if  the  plate  were 
pushed  between  two  flat  boards  held  close  to  each  other. 
Hence  the  following  fundamental  property  of  a  fluid  is 
obtained  from  its  definition  : 

TJie  pressuT-e  of  a  fluid  is  always^tormal  to  any  sur- 
face ivith  which  it  is  in  contact. 

*  Galbraith's  Hydrostatics;  Gregory's  Hydrostatics. 

The  compressibility  of  water  per  atmosphere  at  8°  C,  as  given  in  Everett's 
Units  and  Physical  Constants,  is  48.1  millionths.    Ency.  Brit.,  Vol.  XIL,  p.  439. 


4  MEASURE   OF  PRESSURE. 

5.  Solidifying  a  Fluid.—//  a  mass  of  fluid  be  at 
rest^  any  purtion  of  it  may  he  supposed  to  become 
solid  without  affecting  its  equilibrium  or  the  pressure 
of  the  surroundind  fluid. 

For  there  will  be  no  alteration  in  the  forces  acting  on  the 
fluid,  and  the  action  between  the  solidified  portion  and  the 
rest  of  the  fluid,  or  between  the  solidified  portion  and  any 
surface  with  which  it  may  be  in  contact,  will  still  be  nor- 
mal to  its  surface  (Art.  4) ;  therefore  the  equilibrium  of  the 
solid  can  be  considered  as  maintained  by  the  external  forces 
which  act  upon  it,  and  the  pressure  of  the  remaining  fluid. 

This  proposition  enables  us  to  employ  the  principles  of 
statics  in  the  discussion  of  the  equilibrium  of  fluids. 

6.  Measure  of  the  Pressure  of  Fluids. — The  press- 
ure of  a  fluid  on  a  plane  is  measured,  when  uniform  over 
the  plane,  by  the  force  exerted  on  a  unit  of  area.  Consider 
a  mass  of  fluid  at  rest  under  the  action  of  any  forces, 
and  let  A  be  the  area  of  a  plane  surface  in  contact  with  the 
fluid,  and  P  the  force  which  is  required  to  counterbalance 
the  action  of  the  fluid  upon  A.     Then  if  the  action  of  the 

p 
fluid  upon  A  be  uniform,  -j  is  the  pressure  on  each  unit  of 

the  area  A,  and  this  is  usually  represented  by  p. 

If  the  pressure  be  variable,  as,  for  instance,  on  the  verti- 
cal side  of  a  vessel,  it  must  be  considered  as  varying  con- 
tinuously from  point  to  point  of  the  area  A,  and  the  pressure 
at  any  point  is  measured  by  that  which  would  be  exerted  on 
a  unit  of  area,  supposing  the  pressure  over  the  whole  unit 
to  be  exerted  at  the  same  rate  as  at  the  point  considered. 
If  we  suppose  the  area  A,  and  the  pressure  P,  to  diminish 
indefinitely,  the  pressure  may  be  regarded  as  uniform  on  the 

dP 
infinitesimal  area  dA,  and  we  shall  have  -^^  =i?  to  express 

the  rate  of  pressure  at  the  point  considered. 


PRESSURE   THE  SAME  IN  EVERY  DIRECTION.  5 

By  the  rate  of  pressure  at  a  point  is  meant  the  force 
which  would  be  exerted  on  a  unit  of  area,  if  the  rate  of 
pressure  over  the  unit  were  uniform  and  tlie  same  as  at  the 
point  considered. 

7.  The  Pressure  at  any  Point  of  a  Fluid  at  Rest 
is  the  same  in  every  Direction — By  this  statement  is 
meant  that,  if  at  any  point  of  a  fluid,  there  be  })laced  a  small 
plane  area  containing  the  point,  the  pressure  of  the  fluid 
upon  the  plane  at  that  point  will  bo  independent  of  the  posi- 
tion of  the  plane. 

This  is  the  most  important  of  the  characteristic  properties 
of  a  fluid.  It  is  often  established  by  experiments;  it  may, 
however,  be  deduced,  independently  of  experiments,  in  the 
following  manner : 

Let  a  small  tetrahedron  of  fluid  be  supposed  solidified 
(Art.  5) ;  then  it  is  kept  at  rest  by  the  pressures  on  its  faces, 
which  are  always  normal  (Art.  4),  and  by  the  impressed  * 
forces  on  its  mass.  The  pressures  on  the  faces  depend  on 
the  areas  of  the  faces,  and  the  impressed  forces  depend  on 
the  volume  and  density.  When  the  fluid  is  considered 
homogeneous,  the  former  forces  vary  as  the  square,  and  the 
latter  vary  as  the  cube  of  one  of  the  edges  of  the  solid  ;  sup- 
posing therefore  the  solid  to  be  indefinitely  diminished, 
while  it  always  retains  a  similar  form,  the  latter  forces, 
being  small  quantities  of  the  third  order,  vanish  in  com- 
parison with  the  pressures  on  the  faces, 
which  are  small  quantities  of  the  second 
order;  and  hence  these  pressures  form 
a  system  of  forces  in  equilibrium. 

Let  p,  Pi  be  the  rates  of  pressure 
(Art.  6)  on  the  faces,  ABD,  BCD,  and 
resolve  these  forces  parallel  and  perpen- 
dicular to  the  edge  AC ;  let  (i  and  y  be 

«  See  Anal.  Mechs.,  Art.  234. 


6  HQUAL   TRANSMISSION  OF  PRESSURE. 

the  augles  which  a  plane  perpendicular  to  AC  makes  with 
the  planes,  ABD  and  BCD,  respectively ;  then  we  have 

jo.ABD.cos/3  =j»i. BCD-cosy.  (1) 

But  ABD  •  cos /3  =  BCD -cosy  =  the  projections  of  the 
areas  ABD  and  BCD  on  a  plane  perpendicular  to  AC ; 
therefore  (1)  becomes 

P  =  Pi- 

And  similarly  it  may  be  shown  that  the  pressures  on  the 
other  two  faces  are  each  equal  to  p  or  p^.  As  the  tetrahe- 
dron may  be  taken  with  its  faces  in  any  direction,  it  follows 
that  the  pressure  at  any  point  is  the  same  in  every  direc- 
tion.* 

Cor. — Hence  the  lateral  pressure  of  a  fluid  at  any  point 
is  equal  to  its  perpendicular  pressure. 

ScH. — This  property  constitutes  a  remarkable  distinction 
between  fluids  and  solids,  the  latter  pressing  with  their 
whole  weight  in  the  direction  of  gravity  alone.  This  prop- 
erty of  fluids  can  be  conceived  to  arise  only  from  the  ex- 
treme facility  with  which  the  particles  move  among  one 
another.  It  is  not  easy  to  imagine  how  this  can  take  place, 
if  the  particles  be  supposed  to  be  in  immediate  contact; 
they  are  therefore  probably  kept  at  a  distance  from  one 
another  by  some  repulsive  force. 

8.  Equal  Transmission  of  Fluid  Pressure (1)  Let 

AB  be  a  tube  of  uniform  bore,  and  of  any  shape  whatever, 

filled  with  a  liquid,  and  closed  at  its 

extremities  by  two  pistons  A  and  B, 

which  fit  the  bore  exactly,  but  yet 

can  move  along  it  with  perfect  free-  pj    ^ 

dom ;  and  let  the  interior  of  the  tube 

be  perfectly  smooth,  so  as  not  to  offer  the  least  resistance  to 

*  See  Besant'g  Hydromechanics,  p.  4. 


HQUAL    TBANSMISSION   OF  PRESSURE.  ■         7 

the  motion  of  the  liquid  along  it.  Then  it  may  be  assumed 
as  self-evident,  that  if  any  force  be  applied  to  the  piston  A, 
perpendicular  to  its  surface,  and  directed  inwards,  it  will 
push  the  liquid  forward,  and  thus  produce  a  pressure  on  the 
piston  B,  which  will  drive  it  out  of  the  tube,  unless  there 
be  an  equal  force  at  B,  pushing  in  the  opposite  direction,  to 
countei"act  the  force  at  A  and  keep  the  liquid  at  rest.  This 
property  of  liquids  is  a  direct  result  of  experiment. 

(2)  Let  ABCD  represent  a  closed  vessel  of  any  shape, 
filled  with  a  liquid  ;  let  A  and  B  be  any  two  points  in  the 
surface  of  the  vessel,  and  let  two  circular  holes  be  made  at 
these  points,  having  the  same  area ; 

into  these  let  two  short  tubes  be  in-  

serted,  each  tube  entering  a  little  way     f<w^~-^=-=-^^ -=J-^a 
into  the  liquid,  and  provided  with  a 
piston   that    fits    it    accurately,   and 
which  may  move  within  it  with  the 
utmost  freedom.     Now  suppose  that         D'^^^^^^^^^C 
the  two  orifices,  A  and  B,  are  connect-  '^' 

ed  by  a  tube  of  liquid  AEB,  in  the  interior  of  the  vessel,  of 
uniform  bore,  and  of  any  form,  and  imagine  all  the  liquid 
in  the  vessel,  except  that  contained  in  the  tube,  to  be  solidi- 
fied. This  will  not  affect  the  equilibrium  (Art,  5).  But, 
under  these  circumstances,  if  a  pressure  be  applied  to  the 
piston  A,  and  directed  inwards,  it  will,  as  shown  in  (1) 
above,  be  transmitted  to  B,  and  will  require  an  equal  force 
at  B  to  counteract  it  and  keep  the  fluid  at  rest. 

If  we  suppose  the  piston  B,  to  be  taken  anywhere  on  the 
surface,  it  is  evident  from  what  has  been  said  that  any  press- 
ure applied  to  the  piston  A  will  be  transmitted  to  B,  and 
will  require  an  equal  pressure  at  B  to  counteract  it.  It  is 
also  evident  that  if  we  have  several  openings,  each  equal  to 
B,  closed  by  pistons,  any  pressure  applied  to  one  piston  will 
be  transmitted  undivided  to  every  other  piston,  and  will 
require  an  equal  pressure  at  each  of  those  pistons  to  coun- 
teract it.     The  above  reasoning  remains  true,  no  matter 


8 


EqUILIBRIUM  OF  PRESSURES. 


where  we  suppose  the  point  B  to  be  taken.  Hence  an^ 
pressure,  applied  to  the  surface  of  an  incompressible 
fluid  at  rest,  is  transmitted  equally  to  all  parts  of  the 
fluid  and  to  its  ivhole  surface. 

Cor. — If  a  point  E,  be  within  a  liquid,  the  pressure 
transmitted  from  the  piston  A,  to  a  plane  surface  of  given 
area,  and  having  its  centre  at  E,  is  constant  for  every  pos- 
sible position  of  the  plane,  and  is  always  perpendicular 
to  it. 


9.  The  Pressures  on  Two  Pistons  are  in  Equi- 
librium when  Proportional  to  their  Areas.  —  Let 

Fig.  4  represent  a  vessel  with  two  apertures,  in  which  pis- 
tuns  are  fitted ;  and  let  the  vessel  be 
filled  with  any  liquid.  Now,  any 
pressure  applied  to  the  small  piston  p, 
will  be  transmitted  by  the  liquid  to 
the  large  piston  P,  so  that  every  por- 
tion of  surface  in  the  large  piston 
will  be  pressed  upwards  with  the  same 
force  that  an  equal  portion  of  surface 
in  the  small  piston  is  pressed  down- 
wards (Art.  8).  Let  a  =  the  area  of  the  piston  p,  A  =  the 
area  of  the  piston  P,  p-—  the  whole  pressure  apphed  to  the 
small  piston  JO,  and  P  =  the  whole  pressure  produced  upon 
the  large  piston  P ;  then,  since  the  whole  pressure  on  the 
large  piston  is  equal  to  that  on  the  small  one  taken  as  many 
times  as  the  area  of  the  small  one  is  contained  in  that  of 
the  large,  we  have  for  equilibrium, 


P  =  px-; 
a 


P       A 

or,        —  =  — 
p        a 


(1) 


That  is,  two  forces  applied  to  pistons  which  are  con- 
nected with  each  other  through  the  intervention  of 
some  confined  liquid,   wiU  he  in  equilibrium  when 


EQUILIBRIUM   OF  PRESSURES.  9 

they  are  directly  proportional  to  the  areas  of  the 
pistons  upon  which  they  act. 

This  result  is  wholly  independent  of  the  relative  dimen- 
sions and  positions  of  the  pistons.  Let  a  be  the  unit  of 
area,  say  a  square  inch  or  square  foot,  then  will  p  be  the 
pressure  applied  to  the  unit  of  area,  and  (1)  becomes 

P=pA.  (2) 

That  is,  the  pressure  transmitted  to  any  poHion  of  the 
surface  of  the  vessel  is  equal  to  that  applied  to  the 
unit  of  surface  multiplied  by  the  area,  of  the  surface 
to  which  the  pressure  is  transmitted. 

If  the  area  of  the  piston  P  be  one  square  foot,  and  a 
pressure  of  10  lbs.  is  applied  at  the  piston  p,  it  follows  from 
(2)  that  a  pressure  of  1440  lbs.  will  be  transmitted  to  the 
piston  P,  and  this  must  be  counteracted  by  a  pressure  of 
1440  lbs.  on  that  piston.  Also,  the  interior  of  the  yessel 
will  sustain  an  outward  pressure  of  10  lbs.  on  every  square 
inch  of  its  surface.  And  if  the  pressure  on  the  piston  p,  is 
increased  till  the  vessel  bursts,  the  fracture  is  as  likely  to 
occur  in  some  other  part  as  in  that  towards  which  the  force 
is  directed. 

Cor. — If  in  the  vessel  (Fig.  4)  the  piston  A,  be  made 
sufficiently  large,  the  pressure  transmitted  from  «  to  A  may 
be  increased  indefinitely;  a  very  great  weight  upon  A  may 
be  raised  by  a  small  pressure  at  a,  the  weight  lifted  being 
greater  in  proportion  to  the  size  of  A,  or  inversely  to  the 
size  of  a.  To  increase  the  upward  force  at  A,  we  must 
enlarge  the  surface  of  A  or  diminish  the  surface  of  a,  and 
the  only  limitation  to  the  increase  of  the  force  at  A  will  be 
the  want  of  sufficient  strength  in  the  vessel  to  resist  the 
increased  pressure. 

On  this  principle,  machines  of  immense  mechanical 
power  are  constructed,  which  will  be  described  in  a  future 
chapter. 


10  PRESSURE   OF  A    LIQUID  AT  ANY  DEP-TH. 


EXAMPLES. 

1.  If  the  area  of  the  piston  a  be  a  square  inch,  and  if  it 
be  pressed  by  a  force  of  'Zb  lbs.,  find  the  pressure  which  will 
be  transmitted  to  a  surface  of  35  square  inches. 

Ans.  875  lbs. 

2.  If  the  area  of  the  piston  be  3  square  inches,  and  if  tiie 
pressure  on  it  be  96  lbs.,  find  the  pressure  wliich  will  be 
transmitted  to  a  surface  of  17.5  square  inches. 

Ans.  560  lbs. 

3.  If  the  area  of  the  piston  be  2.5  sq.  in.,  and  if  the  press- 
ure on  it  be  50  lbs.,  what  pressure  will  this  transmit  to  a 
portion  of  the  surface  of  the  vessel  whose  shape  is  circular 
and  whose  diameter  is  one  foot  ?  Ans.  2261.95  lbs. 

10.  Pressure  of  a  Liquid  at  any  Depth.— Thus  far 
only  the  transmission  of  external  pressures  has  been  con- 
sidered ;  we  shall  now  determine  the  effects  of  the  internal 
pressure  due  to  the  weight  of  the  particles  of  the  liquid 
itself. 

Let  DAE  be  the  surface  of  the 
liquid  at  rest,  and  take  any  point  B, 
in  the  liquid ;  draw  B A  vertically  to 
the  surface,  and  describe  a  small  cyl- 
inder about  BA  with  its  base  horizon- 
tal. Imagine  this  cylinder  to  become 
solid  (Art.  5).  Then  this  solid  body 
is  at  rest  under  its  own  weight,  the 
pressure  of  the  fluid  on  the  end  B,  and  '*' 

the  fluid  pressures  on  the  curved  surface. 

The  fluid  pressures  on  the  curved  surface  are  all  horizon- 
tal (Art.  4),  and  the  fluid  pressure  on  the  end  B,  and  the 
weight  of  the  solid  are  vertical  forces,  and  each  group  is 
separately  in  equilibrium.  Hence  the  fluid  pressure  on  B 
must  be  equal  to  the  weight  of  the  solid  AB;  if  a  be  the 


E ^A^ D 


rJiESSUJiJi   OF  A   LIQUID  AT  ANV  DEPTH.  11 

area  of  the  base  AB  z=  z,  w  the  weight  of  a  unit  of  volume, 
and  p  the  pressure  ut  B,  we  have 

pa  =  waz;        or,        p  =^  wz;  (1) 

that  is,  the  pressure  at  any  depth  varies  as  the  depth 
below  the  surface. 

Simihirly,  let  B  and  C  be  any  two  points  in  the  same  ver- 
tical line;  and  let  the  cylinder  BC,  be  solidified;  then, 
from  what  has  just  been  shown,  the  pressures  at  B  and  0 
must  differ  by  the  weight  of  the  cylinder  BC,  i.  e.,  the  press- 
ure at  C  is  greater  than  that  at  B  by  the  weight  of  a  column 
of  licjuid  whose  base  is  equal  to  the  area  0,  and  whose 
height  is  BC. 

Hence,  if  p  and  p'  be  the  pressures  at  B  and  C,  and 
BO  =  z,  we  have 

p'a  —pa  =  waz;        or,        p'  —p  =  toz  ;        (2) 

that  is,  the  difference  of  the  pressures  at  any  two  points 
varies  as  the  vertical  distance  between  the  points. 

Coil.  1. — If  W  be  the  weight  of  a  mass  M,  of  fluid,  then 
(Anal.  Mechs.,  Art.  24),  wo  have 

\y  =  Mg.  (3) 

If  V  be  the  volume  of  the  mass  M,  of  fluid,  and  p  be  its 
density,  then  (Anal.  Mechs.,  Art.  11),  we  have 

M=  Vp.  (4) 

.'.     W  =  gpV.  .       (5) 

For  a  unit  of  volume  we  have  F  =  1,  therefore  (5)  be- 
comes 

W  =  gp. 
From  (1)  we  have, 

pa  =  waz  =  W  =  gp  V  [from  (5)], 


12  PRESSURE   OF  A   LIQUID   AT  ANY  DEPTH. 

or,  pa  =  gpaz  (since  V=az);       (6) 

•••    P  =  gpz-  (7) 

Cor.  2. — If  A  be  the  area  of  the  base  of  a  vessel,  h  its 

height,  and  P  the  whole  pressure  on  the  base,  we  have, 

from  (6), 

P  =  gphA.  (8) 

That  is,  the  pressure  of  a  liquid  on  any  horizontal 
area  is  equal  to  the  weigh  t  of  a  column  of  the  liquid 
whose  base  is  equal  to  the  area,  and  ivhose  height  is 
equal  to  the  height  of  the  surface  of  the  liquid  above 
the  area. 

It  is  evidently  immaterial  whether  the  surface  pressed  is 
that  of  the  base  of  the  vessel  or  a  horizontal  surface  of  an 
immersed  solid. 

Cor.  3. — Since  the  weight  of  a  cubic  foot  of  water  i=  1000 
ozs.  =  62.5  lbs.,  we  have,  for  the  pressure  on  the  bottom 
of  any  vessel  containing  water, 

P  =  62.ohA  lbs.,  (9) 

where  h  is  the  height  in  feet  of  the  surface  of  the  water 
above  the  base,  and  A  the  area  of  the  base  in  square  feet. 

Cor.  4. — The  pressure  on  the  base  of  any  vessel  is 
independent  of  the  form  of  the  vessel. 

Thus,  if  a  hollow  cone,  vertex  upwards,  be  filled  with 
water,  and  if  r  be  the  radius  of  the  base  and  h  the  height 
of  the  cone,  we  have  for  the  pressure  on  the  base, 

P  =  gpTxrVi  [from  (8)], 

or,  P  =  62.5Trr2/A   [from  (9)]  ; 

that  is,  the  pressure  on  the  base  is  the  same  as  if  the  cone 
were  a  cylinder  of  liquid  of  the  same  base  ar.d  height  as  the 


FREE  SURFACE   OF  A   LIQUID  AT  REST.  13 

coDc;  the  pressure  is  three  times  the  weight  of  the  enclosed 
water. 

This  increased  pressure  on  the  base  is  caused  by  the  re- 
action of  the  curved  surface  of  the  cone.  The  pressure  on 
the  curved  surface  consists  of  an  assemblage  of  forces  whose 
vertical  components  all  point  downwards  and  react  upon  the 
base. 

EXAMPLES. 

1.  If  a  surface  of  one  square  inch  be  placed  in  a  vessel 
completely  filled  with  water,  and  if  the  pressure  upon  it  be 
2  lbs.,  what  will  be  the  pressure  on  one  square  inch  placed 
at  a  level  75  inches  lower  ? 

Here  A  =:  one  square  inch,  A  =  75  inches,  and  P  and 
P'  are  the  pressures  at  the  upper  and  lower  points;  there- 
fore we  have,  from  (2)  and  (8), 

P'  —  P  =  252.5*  X  75 

=  18937.5  grains 
=  2.705  lbs. 

.-.     P'  =  2.705  +  2  =  4.705  lbs. 

2.  If  the  pressure  on  the  upper  surface,  whose  area  is  a 
circle  of  half  an  inch  radius,  is  1.5  lbs.,  find  the  pressure  on 
another  circular  area  whose  radius  is  one  inch,  placed  at  a 
depth  10  feet  lower  in  the  water.  Ans.  19.5986  lbs. 

11.  The  Free  Surface  of  a  Liquid  at  Rest  is  a 
Horizontal  Plaue. — Let  ABCD  represent  the  section  of  a 
vessel  containing  a  liquid  subject  to  the 
action  of  gravity;  then  will  its  free 
surface  be  horizontal.  For,  if  the  free 
surface  is  not  horizontal,  suppose  it  to 
be  the  curved  line,  APB.  Take  any 
point  P,  of  the  surface  where  the  tan- 
gent to  the  curve  is  not  horizontal;  let  '^' 

♦  The  weight  of  one  cubic  inch  of  water  at  the  standard  temperature  is  252.5  grains. 


14  FREE  SURFACE   OF  A    LIQUID  AT  REST. 

the  vertical  line  PO,  be  drawn  to  represent  the  weight  ot 
the  particle  of  liquid  at  P,  and  resolve  this  weight  into  two 
components  PR  and  PQ,  the  former  perpendicular,  and  the 
latter  parallel  to  the  surface.  The  first  of  these  is  op})osed 
by  the  reaction  of  the  surface ;  the  second,  being  unopposed, 
causes  the  particle  to  move  downwards  to  a  lower  level.  It 
is  evident,  therefore,  that  if  the  free  surface  be  one  of  equi- 
librium, it  must  at  each  point  be  perpendicular  to  the  direc- 
tion of  gravity,  i.  e.,  it  must  be  horizontal. 

Cor.  1. — Since  the  directions  of  gravity,  acting  on  parti- 
cles remote  from  each  other,  are  convergent  to  the  earth's 
centre,  nearly,  large  surfaces  of  liquids  are  not  plane,  but 
curved,  and  conform  to  the  general  figure  of  the  earth. 
But,  for  small  areas  of  surface  the  curvature  cannot  be  de- 
tected, because  the  deviation  from  a  plane  is  infinitesimal. 

Cor.  2. — The  pressure  of  the  atmosphere  is  found  to  be 
about  14.73  lbs.  to  a  square  inch,  or  very  nearly  15  lbs. 
The  pressure,  therefore,  on  any  given  area  can  be  calculated, 
and  if  rr  be  the  atmospheric  pressure  on  the  unit  of  area, 
the  pressure  at  a  depth  2;  of  a  liquid,  the  surface  of  which 
is  exposed  to  the  pressure  of  the  atmosphere,  will  be,  from 
(7)  of  Art.  10, 

P  =  ffP^-  +  -•  (1) 

Cor.  3. — Since  the  pressures  are  equal  when  the  depths 
are  equal  (Art.  10),  it  follows  that  the  areas  of  equal  press- 
ure are  also  areas  of  equal  depth  ;  therefore,  since  the 
surface  of  a  liquid  is  a  horizontal  plane,  an  area  of  equal 
pressure  is  everywhere  at  the  same  depth  below  a  horizontal 
plane,  i.  e.,  an  area  of  equal  pressure  is  a.  horizontal 
plane  ;  and,  conversely ,  the  pressure  of  a  liquid  at 
rest  at  all  points  of  a  horizontal  plane  is  the  sanme. 

Hence  it  appears  that  when  the  pressure  on  the  surface 
of  a  hquid  is  either  zero  or  is  equal  to  the  constant  atmos- 
pheric pressure,  all  points  on  its  surface  must  be  in  the 


COMMON  SURFACE   OF  TWO   FLUIDS. 


15 


same  horizontal  plane,  even  though  the  continuity  of  the 
surface  be  interrupted  by  the  immersion  of  solid  bodies. 
//  (tinj  number  of  vessels,  containing  the  same 
liquid,  are  in  communication,  the  liquid  stands  at 
the  same  height  in  each  vessel. 

This  sometimes  appears  under  the  form  of  the  assertion 
that  liquids  maintain  their  level. 

Eem. — The  construction  by  which  towns  are  supplied 
with  water  furnishes  a  practical  illustration  of  this  princi- 
ple. Pipes,  leaduig  from  a  reservoir  placed  on  a  height, 
carry  the  water,  underground  or  over  roads,  to  the  tops  of 
houses  or  to  any  point  provided  that  no  portion  of  a  pipe  is 
hisfher  than  the  surface  of  the  water  in  the  reservoir. 


A    a     S     D 


12.  The  Common  Surface  of  Two  Fluids.— Let  AD 

be  the  upper  surface  of  the  lighter  fluid,  and  BC  the  com- 
mon surface  of  the  two  fluids;  AD  is  hori- 
zontal (Art.  11).  Let  P  and  Q  be  two  points 
in  the  heavier  liquid,  both  equally  distant 
from  the  surface  AD,  and  therefore  in  the 
same  horizontal  plane.  Draw  the  vertical 
lines  P«  and  QZ*,  meeting  the  common  sur- 
face of  the  fluids  in  c  and  d.  Let  w  be  the 
weight  of  a  unit  of  volume  of  the  upper  fluid, 
and  lo'  that  of  the  lower. 
Then  we  have 


B 


^Q.  ~=r^  fe 


^e?d3^ 


Fig.  7 


and 


pressure  at  P  =  ?«'-cP  +  w-ac; 
pressure  at  Q  =  w'-dQ,  +  to -id. 


Since  the  pressures  at  P  and  Q  are  eqnal  (Art.  11,  Cor.  3), 
they  being  in  the  same  horizontal  plane,  we  have 

w'-cF  +  w-ac  =  w'-dQ  +  w-bd.  (1) 


But 


cF  -\-  ac  =  f?Q  +  bd 


(^) 


16  TWO   FLUIDS  I.\  A   BENT  TUBE. 

multiplying  (3)  by  w,  and  subtracting  the  result  from  (1), 

we  have 

{w'  —  to)  cP  =  (w'  —  w)  dQ, 

.-.    cP  =  dQ, 

and  hence  BC  is  horizontal. 

That  is,  the  common  surface  of  two  fluids  that  do 
not  mix  is  a  horizontal  plane. 

Cor. — This  proposition  is  true,  whatever  be  the  number 
of  fluids ;  the'  common  surfaces  are  all  horizontal.  If, 
therefore,  the  number  be  infinite,  or  the  density  of  the  fluid 
vary  according  to  any  law,  the  surface  of  eaoh  will  still  be 
horizontal.* 

13.  Two  Fluids  in  a  Bent  Tube Let  A  and  C  be 

the  two  surfaces,  B  the  common  surface. 


^ 


and  p,  p'  the  densities  of  AB  and  BC. 
Let  z  and  z'  represent  the  heights  of 
the  surfaces  A  and  C,  above  the  com- 
mon surface  B,  and  take  B'  in  the 
denser  fluid  in  the  same  horizontal  B 
plane  as  B. 
Then  we  have,  Fig.  8 

the  pressure  at  B   =  gpz  [(7)  of  Art.  10]  ; 

the  pressure  at  B'  =  gp'z\ 

and  these  are  equal  (Art.  11,  Cor.  3). 

.-.    gpz  =  gp'z', 

.:    z:  z'  ::  p' :  p. 

Hence,  when  two  fluids  that  do  not  mix  together 
meet  in  a  bent  tube,  the  heights  of  their  upper  sur- 

*  See  Besant's  Hydrostatics,  p.  31  ;  also  Bland's  Hydrostatics,  p.  20. 


PRESSURE   ON  PLANES. 


17 


faces  above  their  common  surface  are  inversely  pro- 
portional to  their  densities* 

14.  Pressure  on  Planes. — To  find  the  pressure  on 
a  plane  area  in  the  form  of  a  rectangle  when  it  is 
Just  immersed  in  a  liquid,  with  one  edge  in  the 
surface,  and  its  plane  inclined  at  an  angle  6  to  the 
vertical. 

Let  ABCD  be  a  vertical  section  perpendicalar  to  the  plane 
of  the  rectangle ;  then  AB  is  the  section  of  the  surface  of 
the  liquid,  and  AC  {=  a)  is  the 
section  of  the  rectangle,  the  up- 
per edge  b,  of  the  rectangle  being 
in  the  surface  of  the  liquid  per- 
pendicular to  AC  at  A. 

Pass  a  vertical  plane  BC, 
through  the  lower  edge  of  the 
rectangle,  and  suppose  the  fluid 
in  ABC  to  become  solid.  The  weight  of  this  solid  is  sup- 
ported by  the  plane  AC,  since  the  pressure  on  BC  is 
horizontal  (Art.  4).  Let  E  be  the  normal  pressure  on  the 
plane  AC ;  resolving  R  horizontally  and  vertically,  we  have, 
for  vertical  forces, 

Raind  =  weight  of  ABC      ' 

=  gp- ^AB'BC-b    [(5)  of  Art.  10] 
=  l^pa^j  gin  Q  cos  d. 


Fig.  9 


E  =  gpah-\a  cos  6; 


(1) 


that  is,  the  pressure  on  the  rectangle  is  equal  to  the 
iveight  of  a  column  of  fluid  whose  base  is  the  rec- 

*  The  common  barometer  may  be  considered  as  an  example  of  this  principle. 
The  air  and  mercury  are  the  two  fluids.  If  the  atmosphere  had  the  same  density 
throughout  as  at  the  surface  of  the  earth,  its  height  could  be  determined.  For 
height  of  mercury  in  barometpr  :  height  of  air  ::  density  of  air  :  density  of  mer- 
cury. As  mercury  is  10784  times  ;i8  dense  as  air,  the  height  of  the  atmosphere 
would  be  10784  x  30  inches,  or  nearly  5  miles. 


18  THE   WHOLE  PRESSURE. 

tangle,  and  whose  height  is  equal  to  the  depth  of 
the  iniddle  point  of  the  rectangle  below  the  sur- 
face. 

Cor. — When  6  =  0,  (1)  becomes 

R  =  gpah'\a  (2) 

=  gp  (area  BC)  (depth  of  middle  of  BC), 

which  is  the  pressure  on  the  vertical  plane  l&G  ;  hence  the 
law  is  the  same  as  for  the  inclined  plane  AC. 

15.  The  Whole  Pressure. — The  whole  pressure  of 
a  fluid,  on  any  surface  with  which  it  is  in  contact 
is  the  sum  of  the  normal  pressures  on  each  of  its 
elem^ents. 

If  the  surface  is  a  plane,  the  pressure  at  every  point  is 
in  the  same  direction,  and  the  whole  pressure  is  the  same 
as  the  resultant  pressure.  If  it  is  a  curved  surface,  the 
whole  pressure  is  the  arithmetic  sum  of  all  the  pressures 
acting  in  various  directions  over  the  surface.  The  follow- 
ing proposition  is  general,  and  applies  to  curved  or  plane 
surfaces,  for  unit  area. 

Let  8  be  the  surface,  and  p  the  pressure  at  a  point  of  an 
element  dS,  of  the  surface.     Then 

pdS  =  the  pressure  on  the  element ;  (1) 

and  since  the  pressure  is  the  same  in  every  direction  (Art.  7), 
p  will  be  the  normal  pressure  on  this  element,  whatever  be 
its  position  or  inclination.     Hence, 

/    /  pdS  =  the  whole  normal  pressure,  (3) 

the  integration  extending  over  the  whole  of  the  surface 
considered. 


THE    WHOLE  PRESSURE.  19 

If  gravity  be  the  only  force  acting  on  the  fluid,*  we  have, 
from  (7)  of  Art.  10, 

p  =  gpz,  (3) 

z  being  measured  vertically  and  positive  downwards  from 
the  surface  of  the  liquid.     From  (3)  and  (3)  we  have, 

ffpdS=ffgpzd8.  (4) 

Calling  z  the  depth  of  the  centre  of  gravity  of  the  surface 
S,  below  the  surface  of  the  liquid,  we  have  [Anal.  Mechs., 
Art.  84,  (1),  p  and  k  being  constant], 

7-8  =  ffzdS, 
which  in  (4)  gives, 

ffpdS  =  gp'zS,  (5) 

for  the  wliole  pressure  on  the  surface  S.  That  is,  the 
whole  pressure  of  a  liquid  on  any  surface  is  equal 
to  the  weight  of  a  cylindrical  column  of  the  liquid 
whose  base  is  a  plane  area  equal  to  the  area  of  the 
surface  and  whose  height  is  equal  to  the  depth  of 
the  centre  of  gravity  of  the  surface  below  the  sur- 
face of  the  liquid. 

Eem. — The  student  will  now  be  able  to  appreciate  more 
clearly  the  nature  of  fluid  pressures,  and  to  see  that  the 
action  of  a  fluid  does  not  depend  upon  its  quantity,  but 
upon  the  position  and  arrangement  of  its  continuous  por- 
tions. It  must  be  borne  in  mind  that  the  surface  of  an 
incompressible  fluid  or  liquid  is  always  the  horizontal  plane 
drawn  through  the  highest  point  or  points  of  the  fluid,  and 
that  the  pressure  on  any  area  depends  only  on  its  depth 
below  that  horizontal  plane  (Art.  10).  For  instance,  in  the 
construction  of  dock-gates,  or  canal-locks,  it  is  not  the 

*  The  fluid  being  a  homogeneons  liquid. 


20  EXAMPLES. 

expanse  of  sea  outside  which  will  affect  the  pressure,  but  the 
height  of  the  surface  of  the  sea. 

EXAMPLES. 

1.  If  a  cubical  vessel  be  filled  with  a  liquid,  fiud  the  ratio 
of  the  pressures  against  the  bottom  and  one  of  its  sides. 

The  area  of  the  surface  pressed,  in  each  case,  is  the  same, 
but  the  depth  of  the  centre  of  gravity  of  the  bottom  is  twice 
that  of  the  centre  of  gravity  of  the  side  ;  therefore  the  ratio 
is  2  :  1. 

2.  Find  the  pressure  on  the  internal  surface  of  a  sphere 
when  filled  with  water. 

Let  a=  the  radius  of  the  sphere;  then  the  area  of  the 
surface  =  4iTra^,  and  the  depta  of  the  centre  of  gravity  of 
the  surface  below  the  surface  of  the  water  =  a  ;  therefore, 
calling  the  pressure  P,  we  have,  from  (5), 

P  =  gpa-Ana^  =  4^gpna% 

which  is  three  times  the  weight  of  the  water, 

3.  A  rectangle  is  immersed  with  two  opposite  sides  hori- 
zontal, the  upper  one  at  a  depth  c,  and  its  plane  inclined  at 
an  angle  6  to  the  horizontal.  Find  the  whole  pi'essure  on 
the  plane. 

[Let  a  be  the  horizontal  side,  and  b  the  other  side.] 

Ans.  Pressure  =  gpab  Ic  +  -  sin  ^j. 

4.  If  a  cubical  vessel  is  filled  with  water,  and  each  edge 
of  the  vessel  is  10  ft.,  find  the  pressure  on  the  bottom  and 
on  a  side,  a  cubic  foot  of  water  weighing  62^  lbs. 

.        j  Pressure  on  bottom  =  62500  lbs. 
(  Pressure  on  side        =  31250  lbs. 

5.  A  rectangular  surface,  10  ft.  by  5  ft.,  is  immersed  in 
water  with  its  short  sides  horizontal,  the  upper  side  being 


CENTRE   OF  PRESSURE,  21 

20  ft.  and  the  lower  26  ft.  below  the  surface  of  the  water. 
Find  the  pressure  it  sustains.  Ans.  32  tons.* 

6.  A  cylinder,  closed  at  both  ends,  is  immersed  in  a 
liquid  so  that  its  axis  is  inclined  at  an  angle  6,  to  the  hori- 
zon, and  the  highest  point  of  the  cylinder  just  touches  the 
surface  of  the  liquid.  Find  the  whole  pressure  on  the  cyl- 
inder, including  its  plane  ends. 

[Let  r  =  the  radius  of  the  base  and  h  =  the  length  of 
the  cylinder.]         Ans.  gp-^r  {h  -}-  r)  (A  sin  d  -\-  2r  cos  6). 

7.  A  hemispherical  cup  is  filled  with  water,  and  placed 
witli  its  base  vertical.  Find  the  pressures  on  the  curved 
and  plane  surfaces. 

.        \  Pressure  on  the  curved  surface  =  2gpnaK 
\  Pressure  on  the  plane  surface     =:  gp-naK 

This  example  shows  the  distinction  between  the  total 
pressure  of  a  fluid  on  a  curved  surface,  and  on  that  portion 
of  it  which  is  perpendicular  to  any  given  plane.  The  press- 
ure on  the  vertical  plane  side  of  the  hemispherical  cup 
might  be  obtained  by  finding  the  sum  of  the  horizontal 
components  of  the  actual  pressures  on  all  the  elements  of 
the  curved  surface.  This  latter  pressure,  called  the  result- 
atit  horizontal  pressure  of  the  liquid  on  the  surface,  is 
equal  to  the  pressure  of  the  liquid  on  the  plane  base,  other- 
wise the  cup  would  have  a  tendency  to  move  in  a  horizontal 
direction. 

16.  Centre  of  Pressure. — The  centre  of  pressure 
of  a  'plane  area,  immersed  in  a  fluid  is  the  point 
of  action  of  the  resultant  fluid  pressure  upon  the 
plane  area.  It  is  therefore  that  point  in  an  immersed 
plane  surface  or  side  of  a  vessel  containing  a  fluid,  to  which, 
if  a  force  equal  and  opposite  to  the  resultant  of  all  the  press- 

*  One  ton  =  2340  lbs. 


22 


CENTRE   OF  PRESSURE. 


uros  upon  it  be  applied,  this  force  would  keep  the  surface 
at  rest. 

In  the  case  of  a  liquid,  it  is  clear  that  the  centre  of  press- 
ure of  a  horizontal  area,  the  pressure  on  every  point  of 
which  is  the  same,  is  its  centre  of  gravity ;  and  since  the 
pressure  varies  as  the  depth  (Art.  10),  the  centre  of  pressure 
of  any  plane  area,  not  horizontal, 
is  below  its  centre  of  gravity. 

Let  ABCD  be  any  immersed 
plane  area ;  take  the  rectangular 
axes  OX  and  OY,  in  the  plane 
of  the  area.  Let  (.r,  y)  be  any 
point  P,  of  the  area  referred  to 
these  axes,  and  p  the  pressure  at 
this  point,  and  let  EH  be  the  line 
of  intersection  of  the  plane  with 
the  surface  of  the  fluid.  Fig.  lo 

Then  the  pressure  on  the  element  of  area 
=  p  dx  dy  ; 

.*.    the  resultant  pressure  =  J  J  p  dx  dy. 


Let  (^,  y)  be  the  centre  of  pressure ;  then  the  moment  of 
the  resultant  pressure  about  OY 


=1x1  I  pdxdy\ 


and  the  sum  of  the  moments  of  the  pressures  on  all  the  ele- 
ments of  area  about  OY 


=    /  I  pxdx  dy. 


Therefore,  since  the  moment  of  the  resultant  pressure  is 
equal  to  the  sum  of  the  moments  of  the  component  press- 
ures (Anal.  Mechs.,  Art.  59),  we  have 


CENTRE   OF  PRESSURE.  S-3 

x  I  I  p  dxdy  =.    I    I  px  dx  dy  ; 
/  I  pxdxdy 


I  I p dxdy 


(1) 


JJpy  dx  dy 

and,  similarly,  y  =  -jrj- '  (^) 

/    I  p  dx  dy 

the  integration  extending  over  the  whole  of  the  area  con- 
sidered. 

If  polar  co-ordinates  be  used,  a  similar  process  will  give 
the  equations, 

/   /  jyr^  cos  6  dr  dd 

^  =  ^VV >         (^) 

/       pr  dr  dd 

I   I  pr^  sin  Q  dr  dd 
/    /  pr  dr  dd 

If  the  fluid  be  homogeneous  and  incompressible,  and  if 
gravity  be  the  only  force  acting  on  it,  we  have  [Art.  10, 
(7)]. 

jj  =  gpK 

where  /i  (=  PK)  is  the  depth  of  the  point  P  below  the  sur- 
face of  the  fluid,  K  being  the  projection  of  P  on  this  sur- 
face, and  KM  being  perpendicular  to  EH.  Substituting 
this  value  oip  in  (1)  and  (2),  we  get 

I       hxdxdy 
/   j  lidx  dy 


1^  CENTRE   OF  PRESSURE. 

I    fhy  dx  dy 

y  =  "^ ,  (6) 

/    I  Ji  dxdy 

If  we  take  for  the  axis  of  y  the  line  of  intersection  EH, 
of  the  plane  with  the  surface  of  the  fluid,  and  denote  the 
inclination  of  the  plane  to  the  horizon  by  6,  we  have 

PK  =  PM  sin  PMK, 

or,  h  =.  X  sin  0; 

which  in  (5)  and.  (6)  give  us, 

/    I  a^  dx  dy 

I    I  X dxdy 

I   I  xy  dx  dy 

y  =  '^ (8) 

/    I  X dxdy 

Cor.  1. — If  the  axis  of  x  be  taken  so  that  it  will  be  sym- 
metrical with  respect  to  the  immersed  plane,  the  pressures 
on  opposite  sides  of  this  axis  will  obviously  be  equal,  and 
the  centre  of  pressure  will  be  on  this  axis,  or  ^  =  0. 

Cor.  2. — Since  (7)  and  (8)  are  independent  of  6  it  ap- 
pears that  the  centre  of  pressure  is  independent  of  the  incli- 
nation of  the  plane  to  the  horizon,  so  that  if  a  plane  area  be 
immersed  in  a  fluid,  and  then  turned  about  its  line  of  inter- 
section with  the  surface  of  the  fluid  as  a  fixed  axis,  the 
centre  of  pressure  will  remain  unchanged. 

Rem. — The  position  of  the  centre  of  pressure  is  of  great 
importance  in  practical  problems.  It  is  often  necessary  to 
know  the  exact  effect  of  the  pressure  exerted  by  fluids 
against  the  sides  of  vessels  and  obstacles  exposed  to  their 


EXAMPLES.  25 

action,  in  order  to  adjust  the  dimensions  of  the  latter,  so 
that  they  may  be  strong  enough  to  resist  this  pressure. 
Examples  are  furnished  us  in  the  construction  of  reservoirs, 
in  which  large  quantities  of  water  are  collected  and  retained 
for  purposes  of  irrigation,  the  supply  of  cities  and  towns,  or 
to  drive  machinery,  and  of  dykes  to  protect  low  districts 
from  being  inundated  by  seas  and  lakes  and  rivers  in  times 
of  freshets. 

EXAMPLES. 

1.  Find  the  centre  of  pressure  of  a  rectangle  vertically 
immersed,  and  having  one  side  parallel  to  the  surface  of  the 
fluid,  and  at  a  given  distance  below  it. 

Let  a  and  h  be  the  distances  of  the  bottom  and  top  of  the 
rectangle  from  the  surface  of  the  fluid,  and  d  the  width; 
take  the  intersection  of  the  plane  of  the  rectangle  with  the 
surface  of  the  fluid  for  the  axis  of  y,  and  the  middle  point 
of  this  side  for  the  origin,  the  axis  of  x  bisecting  the  rectan- 
gle.    Then  from  (7)  we  have. 


pa  p\d  pa 

j     I    T^dxdy  I  x^dx 

pa    p\d  pa 

I     I   xdxdy  'j  xdx 


,n\d 

J.J 

X  = 


_  2  g«  —  b^ 
—  Za^  —  h^' 

Cor. — If  the  upper  side  of  the  rectangle  is  in  the  surface 
of  the  fluid,  J  =  0,  and  therefore  we  have 

X  =  |«, 

or  the  centre  of  pressure  of  a  vertical  rectangle,  one  side 
being  in  the  surface  of  the  fluid,  is  two-thirds  the  height  Of 
the  rectangle  below  the  surface  of  the  fluid.  The  value  of 
y  is  evidently  zero. 


26  EXAMPLES. 

2.  Find  the  centre  of  pressure  of  an  isosceles  triangle 
whose  base  is  horizontal  and  opposite  vertex  in  the  surface 
of  the  fluid. 

Let  a  be  the  altitude  of  the  triangle  and  h  its  base.  Take 
the  intersection  of  the  plane  of  the  triangle  with  the  surface 
of  the  fluid  for  the  axis  of  y  and  the  vertex  for  the  origin, 
the  axis  of  x  bisecting  the  triangle.     Then  from   (7)  we 

have, 

*-„ 


pa  ffia  pa 

I     I   x^  dxdy         I  a?  dx 

*^o  ^0  t^ 

b  pa 

niia^  I    X^dx 

X  dx  dy        *^o 


3.  A  quadrant  of  a  circle  is  just  immersed  vertically  in  a 
fluid,  with  one  edge  in  the  surface.  Find  its  centre  ot 
pressure. 

Take  the  edge  in  the  surface  for  the  axis  of  y,  and  the 
vertical  edge  for  the  axis  of  x,  and  let  a  be  the  radius. 
Then,  from  (7)  and  (8),  we  have 


pa   p  f'a'— a;"  pa  j 

I     I      x^  dx  dy         I  ix?  {a?  —  x^)^  dx 


and       y  = 


pa   p  Va^ — a;' 
1     1      x  dx  dy 

f^x  (a^  —  x^)i  dx 

a^n       a3         3  . 
16  -^3    =16"^''' 

pa   p  Va^—x' 

1  r^  (a2  -  a^)  dx 

pa   p  Va^—x' 
Jo   Jo       ^'^^^^ 

1  X  {a^  —  a?)^  dx 

8    •  3   ~  8^' 

(See  Besant'g 

5  Hydromechanics,  p.  41.) 

EMBANKMENT    WHEN   ONE  FACE  IS    VERTICAL.        27 

4.  Find  the  centre  of  pressure  of  the  triangle  in  Ex,  2, 
when  it  is  inverted  so  that  the  base  is  in  the  surface  of  the 
fluid. 

Ans.  At  a  distance  of  ^a  below  the  surface  of  the  fluid. 

5.  An  immersed  rectangle  has  two  sides  horizontal,  the 
inclination  of  the  plane  of  the  rectangle  to  the  horizon  is 
d,  the  depth  of  its  upper  side  below  the  surface  of  the  fluid 
is  Cf  the  sides  of  the  rectangle  are  a  and  b,  the  latter  hori- 
zontal.    Find  its  centre  of  pressure. 

[Take  the  upper  side  for  the  axis  of  y,  and  its  middle 
point  for  the  origin.] 

.        _       a  3c  +  2a  sin  0        ,  _ 

Ans.  X  =  ^'— ^-—   and  «  =  0. 

3    2c  +  a  sin  6  ^ 

17.  Embankments. — An  embankment  generally  con- 
sists of  a  large  mass  of  earth  and  other  material.  When 
used  for  the  side  of  a  reservoir  or  canal,  to  bank  up  a  river,* 
to  keep  out  the  sea,f  or  in  general  to  dam  back  water,  they 
are  constructed  on  certain  principles,  and  are  opposed  to  the 
effort  made  by  the  water  to  spread  itself.  The  effort  to 
overthrow  the  embankment  arises  from  the  force  which  the 
water  exerts  horizontally  ;  and  the  stability  is  caused  by  the 
weight  of  the  embankment.  When  therefore  there  is  an 
equilibrium,  the  former  of  these  forces  must  be  equivalent 
to  the  latter. 

An  embankment  is  generally  made  wider  than  is  abso- 
lutely necessary,  to  give  strength  and  stability  sufficient  to 
insure  it  against  all  risks.  Frequently  they  slant  only  on 
the  side  that  is  away  from  the  water.  In  every  case  the 
embankment  should  be  built  much  stronger  at  the  bottom 
than  at  the  top,  for  the  pressure  of  water  increases  as  the 
depth. 

18.  Embankment  when  the  Face  on  the  Water 
Side  is  Vertical. — Find  the  stability  of  an  embankment 

*  Called  dykes.  t  Called  eeawalls. 


28        EMBANKMENT   WHEN  ONE  FACE  IS    VERTICAL. 


^mp 


whose  section  has  the  form  of  a  trapezoid  when  the  water 
stands  at  its  brim. 

Let  ABCD  be  the  cross-section  of  the  embankment ; 
draw  DE  parallel  to  the  vertical  side  BC ;  let  G  and  g  be 
the  centres  of  gravity  of  the  rectangle 
and  triangle  respectively;  draw  the 
vertical  lines  GH  and  ^K  ;  let  A.B 
=  a,  DC  =  b,  BC  =  h,  iv  =  the 
weight  of  each  cubic  foot  of  the  ma- 
terial, and  Wj  ^  the  weight  of  a 
cubic  foot  of  water. 

The  forces  acting  are  the  weight 
of  the  wall,  and  the  fluid  pressure  on 
BC.  As  the  embankment  is  uniform 
throughout  its  length,  and  also  the  pressure  on  it,  we  may 
determine  the  stability  by  taking  only  one  foot  in  length. 
Take  BM  =  |BC,  and  M  will  be  the  centre  of  pressure 
(Art.  16,  Ex.  1,  Cor.),  The  resultant  P,  of  the  pressure  of 
the  water  against  the  wall  acts  at  M,  and  tends  to  turn  the 
embankment  over  its  outer  edge  A.     Hence,  we  have 

the  moment  of  P  ^  pressure  of  water  on  BC  x  AO 

(Art.  15) 
=  yi^w^  X^h  =  ^h^Wi  ;  (1) 

the  moment  of  AED  =  weight  of  AED  x  AK 
=  ^{a—b)hwx^{a—b) 
=  ^{a  —  b)''^7iw; 
the  moment  of  EBCD  =  weight  of  EBCD  x  AH 
=  bhw  X  (a — ^b) ; 
.-.   the  moment  of  ABCD  =  [^(a—iy  +  b{a—^b)]Jiw.  (4) 

If  the  embankment  be  upon  the  point  of  overturning  on 
A,  the  moments  in  (1)  and  (4)  are  equal  to  each  other,  and 
we  have 


(3) 


(3) 


EMBANKMENT   WHEN  ONE  FACE  IS  SLANTING.       29 

yi^tc^  =  [i(«-5)2  +  b{a—^b)]hw, 
or,  P  =  [2  {a-bf  -H  35  (2«-6)]  — ,  (5) 

and  the  embankment  will  be  overturned  or  not,  according  as 
h  >  or  < 


Y/[2(«-5)2  +  3*(2a-&)]|^. 


Coil. — If  the  embankment  is  rectangular,  h  ^=  a,  and  (5) 
becomes 

h^  =  ^a^^-  (6) 

If  the  embankment  is  triangular,  J  =  0,  and  (5)  becomes 
7.2  =  2«^^. 


19.  Einbaukment  when  the  Face  on  the  Water 
Side  is  Slaiitiiig. — Find  the  stability  of  an  embankment 
whose  section    is    a   trapezoid   which 
slants  on  both  sides,  viz.,  towards  the 
water  and  away  from  it. 

(1)  Suppose  the  embankment  to 
yield  to  the  pressure  of  the  fluid  by 
turning  round  the  outer  edge  A. 

Let  ABCD  be  the  cross-section  of 
the  embankment.  Since  the  pressure 
of  a  fluid  is  always  normal  to  the  sur- 
face with  which  it  is  in  contact  (Art. 
4),  the  pressure  on  the  slanting  face  BC,  of  this  embank- 
ment is  inclined  to  the  horizon,  and  hence  the  stability  of 
the  embankment  is  caused  by  its  weight  and  the  vertical 
pressure  of  the  fluid  on  the  face  BC,  while  the  effort  to 
overthrow  it  is  caused  by  the  horizontal  pressure  of  the 
fluid. 

Let  P ^  and  P^  be  the  horizontal  and  vertical  components 


30        EMBANKMENT   WHEN   ONE  FACE  IS  SLANTING. 

of  the  normal  pressure  P,  and  «  the  angle  which  the  direc- 
tion of  the  normal  pressure  makes  with  the  horizon  ;  then 
we  have,  for  the  horizontal  component, 

Pj  =  P  cos  a 

=  area  of  BC  x  |CE  x  lo^^  cos  «  (Art.  15) 
=  area  of  CE  x  Ihw^ , 

where  h  —  OE,  and  w^  is  the  weight  of  a  cubic  foot  of  the 
fluid. 

Similarly,        P^  =  area  of  BE  x  \luo^ ; 

but  area  of  CE  is  the  projection  of  BC  on  CE,  and  area  of 
BE  is  the  projection  of  CB  on  EB  ;  i.  e.,  the  pressure 
exerted  by  a  fluid  in  any  direction  upon  a  surface 
is  equal  to  the  weight  of  a  column  of  the  fluid, 
whose  base  is  the  projection  of  the  surface  at  right 
angles  to  the  given  direction,  and  whose  height  is 
the  depth  of  the  centre  of  gravity  of  the  surface 
below  the  surface  of  the  fluid. 

Hence,  since  the  projection  at  right  angles  to  the  vertical 
direction  is  the  horizontal  projection,  and  that  at  right 
angles  to  a  horizontal  direction  is  a  vertical  one,  we  find 
the  vertical  pressure  of  the  fluid  against  a  surface  by  treat- 
ing its  Jiorizonlal  projection  as  the  surface  pressed  upon,  and, 
on  the  contrary,  the  horizontal  pressure  of  the  fluid  in  any 
direction  by  treating  the  vertical  projection  of  the  surface  at 
right  angles  to  the  given  direction  as  the  surface  pressed 
upon,  and  in  both  cases  we  must  regard  the  depth  of  the 
centre  of  gravity  of  the  surface  below  tlie  surface  of  the 
fluid  as  the  "  height  of  the  column." 

Let  g,  G,  and  g^  be  the  centres  of  gravity  of  AFD, 
FECD,  and  EBC ;  let  AB  =  »,  DC  =  h,  AF  =  c,  EB  =  d, 
and  10  =  the  weight  of  each  cubic  foot  of  the  embankment. 
The  horizontal  pressure  of  the  water  acting  at  M  tends  to 


EMBANKMENT   WHEN  ONE  FACE  IS  SLANTTNG.       31 

turn  the  embankment  over  its  outer  edge  A.     Ilence,  we 
have 

the  moment  of  P^  =  yi^w^  x^h  =  ^hhv^ ;  (1) 

the  moment  of  Pg  =  <^  X  ^hic\  x  AH 
=  \hdw^  {a  —  ^d) ; 

the  moment  of  AFD  =  wt.  of  ADF  x  |AF 

=  ^htv  X  f c  =  \cViw ; 

tho.  moment  of  FECD  =  wt.  of  FECD  x  (AF  + JFE) 

=  hlno  X  (c  +  \h) ; 

the  moment  of  EBC  =  wt.  of  EBC  x  (AB  —  |BE) 
=  \dliw  X  (a  —  |rf) 
=  \dJiw  (3a  —  2(/). 

.-.  the  moment  of  ABCD  =  ^  +  lj(2c+*) 

+  -  (3a  —  M)  \hw-\--  (3a  —  d)  hw^.        (ii) 

If  the  embankment  be  upon  tlie  point  of  overturning  on 
A,  the  moments  in  (1)  and  (2)  are  equal  to  each  other,  and 
we  have 


^h^w^=     ^  +  |(2c  +  Z,)+^(3a-2^) 


hw  +  ~  {3a— d)  hwi 


w 
or,   7*2=  [2(^-{-3b{2c  +  b)-\-d{3a-2d)]—-\-diSa—d).{3) 

and  the  embankment  will  be  overturned  or  not,  according  as 

h  >  or  <A/[2c2  +  3*(2c  +  *)  +  f/(3a— 2(Z)]  — +^(3a-f?). 

Cor.  1. — If  the  embankment  is  of  the  form  of  Fig.  11, 
(Z  =  0,  and  (3)  becomes 


32        EMBANKMENT   WHEN  ONE  PAC£  IS  SLANTING. 

h^=  [2c^^3b{2c  +  b)]^,  (4) 

which  agrees  with  (5)  of  Art.  18. 

CoK.  2. — If  the  embankment  is  rectangular,  c  =  0,  and 

(4)  becomes 

w 
h^  =  362  —  , 

which  agrees  with  (6)  of  Art.  18. 

(2)  Suppose  the  embankment  to  yield  to  the  pressure  of 
the  fluid  by  sliding  along  the  horizontal  base  AB. 
The  horizontal  pressure  of  the  fluid,  from  (1),  is 

p,  =  W^^i ; 

the  vertical  pressure  of  the  fluid  is 

Pg  =  l-dhw^. 
The  weight  of  the  embankment  is 
a  +  b 


2 


hw ; 


and  the  entire  vertical  pressure  of  the  embankment  and  the 

water  on  its  face  is 

a  -\-b 

— - —  hio  +  ^amoi 

=  {aw  -{-  bw  +  dwi)  ^?i. 

Let  ju  =  the  coefficient  of  friction  ;  then  the  friction 
between  the  embankment  and  the  surface  of  the  ground  on 
which  it  rests  is  (Anal.  Meclis.,  Art.  92), 

{mv  +  ho  4-  dw^)  yifi. 

When  the  horizontal  pressure  of  the  water  pushes  the 
embankment  forward,  we  must  have 

\]i:ho^  =  {aw  +  iw  +  dtu^)  ^hfj. ; 


PRESSURE    UPON  BOTH  SIDES   OF  A   SURFACE.        33 


or,  more  simply,    h  = 


{a  +  *) 


i^  +  d"' 


(8) 


and  the  dam  will  move  or  not  according  as 


A  >  or  < 


tv 


(a  +  b)—  -}-d 


to. 


Cor. — If   the  embankment  is  rectangular,    d  =  0   and 
h  =  a,  and  (5)  becomes 


h  =  2a  I-  /x. 


20.  Pressure  upon  Both  Sides  of  a  Surface.— If 

a  plane  surface  is  subjected  on  both  sides  to  the  pressure  of 
a  fluid,  the  two  resultants  of  the  pressures  on  the  two  sides 
have  a  new  resultant,  which,  as  they  act  in  opposite  direc- 
tions, is  obtained  by  subtracting  one  from  the  other. 

Let  AB  be  a  flood-gate  with  the 
water  pressing  on  both  sides  of  it,  to 
determine  the  resultant  pressure,  and 
the  centre  of  pressure.  Let  AB  =  «, 
the  depth  of  the  water  on  one  side ; 
DB  =  b,  the  depth  of  the  water  on 
the  other  side  ;  P  =  the  resulting 
pressure  on  the  gate  ;  and  Wj  =  the 
weight  of  a  cubic  foot  of  water.    Then 

F  =  pressure  on  AB  —  pressure  on  DB  ; 

.-.         P  =  ^  («2  _  J2)  to^.  (1) 

Now  let  C  and  Cj  be  the  centres  of  pressure  of  the  sur- 
faces AB  and  DB,  and  Cg  the  point  to  which  the  resultant 
pressure  P,  is  applied.  Then,  taking  moments  with  respect 
to  A,  and  putting  ACg  =  "z,  we  have 

P  xi  =  pressure  on  AB  x  AC  —  pressure  on  DB  x  AC, 
=  ^cfiw^  X  f«  —  ^li^iVi  {a  —  -JJ). 


34  EXAMPLES. 


'  =  -^J^2^r^~   l^'om  (1)] 
2a2  +  2aJ  —  *« 


3  {a  +  A) 


(2) 


EXAMPLES, 


1.  The  total  breadth  of  a  flood-gate  is  2b  feet,  and  the 
depth  is  a  feet ;  the  hinges  are  placed  at  d  feet  from  the 
respective  extremities  of  the  gate ;  required  the  pressure 
upon  the  lower  hinge. 

Let  AB  represent  the  height  of  the  gate,  j^?^^^' 

D  and  E  the  hinges,  and  0  the  centre  of  ^J^^jq 
pressure  of  the  water.     The  pressure  of  the 
water  upon  each  half  of  the  gate  =  ^a^w  ; 
and  since  the  pressure  of  the  water  at  0  is 

supported  by  the  hinges  D  and  E,  we  have,  ^^^^-^ 

by  the   equality  of  moments   with   respect  ^^'"^^"=^ 

toD,                ^  '^'«-" 

Pressure  on  E  x  DE  =  Pressure  on  C  x  DC ; 

but  DE  =  a  —  2d,     and     DC  =  f  a  —  <f  ; 

.*.     Pressure  on  E  {a  —  2d)  =  ^a^biv  {\a  —  d) ; 

„                   „       ci^bw  (2a  —  Zd) 
.'.     Pressure  on  E  =  — —-r ttt — -' 

6  (a  —  2d) 

2.  A  brick  wall,  with  rectangular  cross-section,  12   ft 
high  and  3  ft.  thick,  sustains  the  pressure  of  water  against 
one  of  its  faces.     Find  the  height  to  which  the  water  may 
rise  without  overthrowing  the  wall,  each  cubic  foot  of  the 
wall  weighing  112  lbs. 

Ans.  8.34  ft.,  or  within  3.66  ft.  of  the  top  of  the  wall. 

3.  A  brick  wall,  whose  cross-section  is  a  right-angled  tri- 
angle, weighs  120  lbs.  per  cubic  foot,  and  sustains  the 
pressure  of  water  against  its  vertical  face;   its  height  is 


ROTATING   LIQUID.  35 

14  ft.,  and  its  base  is  6  feet.  Show  that  the  wall  will  be 
overthrown  by  the  pressure  of  water  against  it,  when  it  rises 
to  the  top  of  the  wall. 

21.  Rotating  Liquid. — It  has  been  shown  (Art.  11) 
that,  if  a  liquid  at  rest  be  subject  to  the  force  of  gravity 
only,  its  free  surface  must  be  horizontal,  i.  e.,  everywhere 
perpendicular  to  the  direction  of  gravity.  In  the  same  way 
it  may  be  shown  that,  if  a  liquid  be  subject  to  any  forces 
whatever,  its  surface,  if  free,  must  at  every  point  be  per- 
pendicular to  the  resultant  of  the  forces  which  act  upon 
that  point.  For,  if  the  resultant  had  any  other  direction, 
it  could  be  resolved  into  two  components,  one  in  the  direc- 
tion of  the  normal  and  the  other  in  the  direction  of  the 
tangent;  the  first  of  these  would  be  opposed  by  the  reac- 
tion of  the  surface  ;  the  second,  being  unopposed,  would 
cause  the  particle  to  move,  which  is  contrary  to  the  hypoth- 
esis that  the  surface  is  at  rest:  hence  the  surface  is  at  every 
point  perpendicular  to  the  resultant  of  the  forces  which  act 
upon  that  point. 

If  a  quantity  of  liquid  in  a  vessel  he  made  to  rotate 
uniformly  about  a  vertical  axis,  the  surface  of  the 
liquid  will  taJce  the  form,  of  a  paraboloid  of  revolu- 
tion. 

Let  ABCD  represent  a  vertical  section 
made  by  a  plane  passing  through  ZZ',  the 
axis  of  rotation  of  the  vessel  containing 
the  liquid,  and  let  the  curved  line  AVD, 
represent  the  section  of  the  surface  of 
liquid  made  by  this  plane,  and  let  P  be 
any  point  taken  on  this  section. 

Now  every  particle  of  the  liquid  moves 
uniformly  in  a  horizontal  circle  whose 
centre  is  in  the  axis  ZZ',  and  there- 
fore is  urged  horizontally  by  a  centrifugal  force  directed 


36  ROTATING   LIQUID. 

from  the  axis.  Let  m  be  the  mass  of  the  particle  at  P,  u 
the  angular  velocity  of  the  liquid,  and  y  the  distance  MP, 
and  denote  the  centrifugal  force  by  P;  then  (Anal.  Mechs., 
Art.  198)  we  have,  for  the  centrifugal  force  on  the  parti- 
cle m, 

P  =   W?u)2y.  (1) 

The  particle  is  also  urged  vertically  downwards  by  its 
own  weight  mg,  due  to  the  force  of  gravity ;  hence  the  par- 
ticle is  in  equilibrium  under  the  action  of  gravity  mg,  of 
the  centrifugal  force  moj^y,  and  of  the  reaction  of  tiie  sur- 
face of  the  liquid  which  is  normal,  and  therefore  the  result- 
ant of  mg  and  7no)^y  must  be  normal  to  the  surface. 

Let  PF  and  PG  represent  the  centrifugal  force  and  force 
of  gravity,  respectively  ;  then,  completing  the  parallelogram 
offerees,  the  resultant  of  these  PR,  must  be  normal  to  the 
surface  at  P.  Let  this  normal  meet  the  axis  in  N;  since 
the  triangles,  GPR  and  MNP,  are  similar,  we  have 

NM  :  MP    ::    PG  :  GR  (=  PF); 

or  NM  :     y      ::    mg  :  mto^y ; 

.-.    NM=5.  (2) 

But  NM  is  the  subnormal  of  the  curve,  AVD  ;  therefore 

the  subnormal  NM  =z  ^  =  a  constant, 

which  is  a  property  of  the  parabola.  Hence  the  curve 
AVD,  is  a  parabola  whose  latus  rectum  is  -^,  and  therefore 
the  surface  is  a  paraboloid  of  revolution. 

ScH. — It  will  be  seen  that  this  result  is  independent  of 
the  form  of  the  containing  vessel.  The  axis  of  rotation,  in 
fact,  may  be  within  or  without  the  fluid,  but  in  any  case  it 
will  be  the  axis  of  the  surface  of  the  pai'aboloid. 


PRESSURE  AT  ANY  POINT  OF  A   ROTATING   LIQVID.    B7 


EXAMPLES. 

.     1.  If  the  vessel  (Fig.  15)  contain  a  liquid,  and  make  30 
revolutions  per  minute,  find  the  value  of  NM. 

Here  w  =  2::  x  30-^-60  =  tt,  and  ^  =-  33  ;   therefore  we 
have,  from  (2), 

NM  =  ~  =  3.242  ft.  =  38.9  in. 

2.  If  the  vessel  make  one  turn  in  a  second,  find  the  value 
of  NM.  Ans.  9.72  in. 

3.  If  the  vessel  make  95  turns  per  minute,  find  the  value 
ofNM.  Ans.  3.88  in. 

22.  The  Pressure  at  any  Point  of  a  Rotating 
Liquid. — Let  ABCD  be  a  vertical  section  through  the  axis 
of  a  vessel  containing  a  rotating 
liquid ;  let  Q  be  any  point  {x,  y)  in 
the  liquid  referred  to  the  rectangu- 
lar axes  OX,  OY,  and  describe  a 
small  vertical  prism  having  Q  in  its 
base,  which  is  to  be  taken  hori- 
zontal. 

The  prism  PQ  of  liquid  rotates 
uniformly  under  the  action  of  the 
pressure  around  it,  but  its  weight  is 
entirely  supported   by   the   vertical   pressure  on   its  base. 
Hence,  if  ^  be  the  pressure,  and  p  the  density,  we  have 


Fig.  1.6 


Bat  [Art.  21,  (2)], 
PQ  = 

which  in  (1)  gives. 


P  =  ffP^Q- 


PQ  =  OM  -  ON  =  "^^^  -  ON, 


(1) 


p  —  p  (i'^^^  —  9y)> 


(2) 


98 


EXAMPLES. 


which  gives  the  pressure  at  any  point  in  terms  of  tiie  an^ni- 
lar  velocity  and  of  the  co-ordinates  of  the  jjoint  referred  to 
the  axis  and  vertex  of  tlie  paraboloid.  (See  Besant's  Hy-^ 
drostatics,  p,  152.) 

Cor. — If  Q  be  lower  than  0,  y  is   negative,  and  (3) 
becomes 

P  =  p{^<^^^^  +  <Jh)'  (3) 

EXAMPLES. 

1.  A  tube  ABCD,  the  equal  branches  of  which  are  verti- 
cal, and  BC  horizontal,  is  filled  Avith  liquid  and  made  to 
rotate  uniformly  about  the  axis  of  AB ; 
find  how  much  liquid  will  flow  out  of  the 
endD. 

The  liquid  will  flow  out  until  the  sur- 
face in  AB  is  the  vertex  of  a  parabola 
passing  through  D,  and  having  its  axis 

vertical  and  latus  rectum  =  ~  (Art.  21). 

If  then  0  be  the  vertex  of  the  parabola, 
we  shall  have 

AD^  =  ^,AO; 


.',    AO  =  ^AD^ 

which  gives  AO,  and  thus  determines  the  quantity  which 
flows  out. 

If,  however,  AO  be  greater  than  AB,  i.  e.,  if  G  be  ht^ow 
B,  at  0',  for  instance,  the  surface  of  tbe  liquid  will  be  in 
BC,  at  P.     We  shall  then  have, 

AD'  =  %KO'i 


STRENGTH  OF  PIRES  AND  BOILERS. 


39 


and 


gp2^  ?^B0'; 


AD' 


which  determines  the  position  of  P.     (Besant's  Hydrostatics, 
p.  154.) 

2.  A  straight  tube  AB,  filled  with  liquid,  is  made  to 
rotate  about  a  vertical  axis  through  A ;  find  i,,n 

how  much  flows  out  at  B. 

Ans.  All  above  P,  where  P  is  tangent  to 

2^ 
the  parabola  whose  latus  rectum  is  ~  and 

whose  axis  is  coincident  with   the  vertical     0-' 

line  through  A,  and  AP  =  —  cot  «  cosec  «, 


where  «  is  the  angle  OAB. 


Fig.  18 


23.  Strength  of  Pipes  and  Boilers. — An  important 
application  of  the  theory  of  the  pressure  of  fluids  is  the 
determination  of  the  thicliness  of  pipes,  boilers,  etc.  In 
order  tiiat  these  vessels  shall  be  strong  enough  to  resist  the 
pressure  of  the  liquid,  their  rvalls  must  be  made  of  a  certain 
thickness,  which  depends  upon  the  pressure  of  the  liquid 
and  the  internal  diameter  of  the  vessel. 

Let  it  he  required  to  find  the  thickness  of  a  pipe  of 
any  material  necessary  to  resist  a  given  pressure. 

A  cylindrical  vessel  may  burst  either  transversely  or  lon- 
gitudinally ;  but  the  former  is  less  likely  to  occur  than  the 
latter,  as  appears  from  the  following  investigation. 

(1)    JV7iefi  the  rupture  is  transverse. 

Let  A  BCD  (Fig.  19)  be  a  section  of  pipe  perpendicular 
to  its  axis,  the  interior  surface  of  which  is  siibjected  to  a 


40 


STRENGTH  OF  PIPES  AND   BOILERS. 


pressure  oi p  on  each  unit  of  surface.  Let  2r  be  the  diam- 
eter MD  of  the  interior,  then  will  the  surface  pressed  be 
measured  by  -nr^,  which  is  the  area  of  the 
cross-section  of  the  interior,  and  the 
whole  pressure  upon  the  surface  of  the 
end  of  the  pipe  and  which  produces  rup- 
ture will  be  measured  by 


Trr^p. 


(1) 


Let  e  =  AE  =  the  thickness  of  the 
pipe ;  then  the  cross-section  of  the  mate- 
rial of  the  pi})e 

=  TT  (r  +  ey  —  Trr2  =  ne  {e  +  2r). 

Let  T  denote  the  strength  of  the  material  of  which  the 
pipe  is  composed,  for  each  unit  of  cross-section ;  then  the 
strength  of  the  entire  pipe  in  the  direction  of  the  axis 

=  -e{e  +  2r)  T,  (2) 

and  since  the  whole  pressure  in  (1)  when  rupture  is  about 
to  take  place  must  be  held  in  equilibrium  by  the  strength 
in  (3),  we  have 

ne  {e  +  2r)  T  =  nr^p, 


rp 


rp 
2l" 


K^  +  l)^ 

since  e  is  usually  very  small  in  comparison  with  2r. 


(3) 


(2)    When  the  rapture  is  longitudinal. 

Let  EMH  be  any  portion  of  the  wall  whose  length  is  I, 
and  let  2«  =  the  angle  ECH.  Then,  since  the  projection 
of  EMH  at  right  angles  to  the  line  MD  passing  through 
the  centre  is  a  rectangle  whose  area  =i  2rl  sin  «,  the  mean 
pressure  of  the  fluid  on  the  wall,  EMH 

—  2r?sin  «j5  (Art.  19).  (4) 


STRENGTH  OF  PIPES  AXD   BOILERS.  41 

Now  this  pressure  must  be  held  in  equilibrium  by  the 
forces  of  cohesion,  E,  R,  acting  tangentially  on  the  cross- 
sections,  AE  and  BH,  of  the  wall  of  the  pipe.  Denoting 
the  components  of  R,  R,  parallel  to  MD,  by  Q,  Q,  we  have 

2^  =  2R  sin  «  =  2elT sm  a,  (5) 

e  being  the  thickness  of  the  pipe  and  T  the  strength  of  eacli 
unit  of  section. 

Therefore,  from  (4)  and  (5)  we  have, 

2elTsm  a  =  2rlp  sin  «; 

•••    .  =  f ,  (6) 

which  shows  that  the  thickness  of  the  pipe  is  independent 
of  its  length. 

Otherwise  thus,  hy  the  principle  of  ivorh. 

The  whole  surface  of  the  interior  of  the  pipe  =  2-nrl ;  and 
the  whole  pressure  upon  the  surface  =  %-nrlp.  Suppose  the 
pipe  to  rupture  longitudinally,*  under  this  pressure,  its 
radius  becoming  r-\-dr',  then  the  path  described  by  the 
pressure  will  be  dr,  and  the  work  done  by  the  pressure 

=  2TTrlp  dr.  (7) 

The  force  R,  which  resists  rupture  and  acts  tangentially, 
=  eTl.  While  the  radius  of  the  interior  changes  from  r  to 
r  +  dr,  the  circumference  changes  from  27rr  io  ^ir  {r -{■  dr) ', 
then  the  path  described  by  the  resistance  =  2,tc  dr,  and  the 
work  done  by  the  resistance 

=  )lTTeTl  dr.  -  (8) 


*  Longitudinal  tension  produces  transverse  rupture,  and  transverse  tension  pro- 
duces longitudinal  rupture.  The  stretching  tendency  to  rupture  longitudinally  is  a 
transverse  stretching,  i.  e.,  the  pipe  tends  to  bulge  out  all  along  its  length  ;  hence, 
transversely,  r  becomes  r+dr. 


42  EXAMPLES. 

Therefore,  from  (7j  and  (8),  by  the  principle  of  work,  we 
have 

2-neTl  (Ir  =  %-nrl])  dr. 


"   —    rp  f 


which  is  the  same  as  (6). 


From  (3)  and  (6)  it  follows  that,  to  prevent  a  longi- 
tudinal rupture,  the  wall  must  he  made  twice  as 
thick  as  would  be  necessary  to  prevent  a  transverse 
one. 

COK.— Since  p  =  zw  [from  (1)  of  Art.  10],  (3)  and  (6) 
become,  respectively, 

rp  rziv  ,  rp  rzw 

^  —  22'  ~  W'  ^  —  -f  —  "T' 

that  is,  the  thickness  of  similar  pipes  must  vary  di- 
rectly as  their  diameter  and  as  the  pressure  upon  the 
unit  of  surface,  or  in  the  case  of  a  liquid,  as  the 
depth  of  the  pipe  below  the  upper  surface  of  the 
liquid,  and  inversehj  as  the  strength  of  eacJb  unit  of 
section. 

A  pipe  which  has  twice  the  diameter,  and  has  to  sustain 
four  times  the  pressure  of  another,  must  be  eight  times  as 
thick.  (See  Weisbach's  Mechs.,  Vol.  L,  p.  739;  Bartlett's 
Mechs.,  p.  294;  Tate's  Mechs.,  p.  268.) 

EXAMPLES. 

1.  It  is  found  that  the  pressure  is  uniform  over  a  square 
yard  of  a  plane  area  in  contact  with  fluid,  and  that  the 
pressure  on  the  area  is  13608  lbs.;  find  the  measure  of  the 
pressure  at  any  point  (Art.  6),  (1)  when  the  unit  of  length 
is  an  inch,  (2)  when  it  is  two  inches. 

Ans.  (1)  10|  lbs. ;  (2)  42  lbs. 


EXAMPLES.  43 

3.  If  the  area  of  a  (Fig.  4)  be  a  square  inch,  and  if  it  be 
pressed  by  a  force  of  15  lbs.,  what  pressure  *  will  this  trans- 
mit to  the  piston  A  if  its  diameter  be  10  in.  ? 

Ans.  Pressure  on  A  =  1178  lbs. 

3.  If  the  diameter  of  a  be  4  in.,  and  if  the  pressure  on  it 
be  185  lbs.,  what  pressure  will  be  exerted  on  A  if  its  area  is 
one  square  foot?  Ans.  Pressure  on  A  =  2120  lbs. 

4.  If  the  area  of  a  be  20  square  inches,  and  if  it  be 
pressed  by  a  force  of  3G0  lbs.,  find  the  diameter  of  A  so 
that  it  shall  be  pressed  upwards  by  a  force  of  10  tons  (one 
ton  =  2240  lbs.)  Ans.  Diameter  of  A  =  39.8  in. 

5.  If  the  diameter  of  A  (Fig.  3)  be  one  inch,  and  if  the 
surface  at  E  be  a  square  whose  side  is  one-quarter  of  an 
inch,  find  the  pressure  transmitted  to  E  if  that  on  A  be 
10  lbs.  Ans.  Pressure  on  E  =  0.795  lbs. 

6.  If  the  area  of  A  be  2^  sq.  in.,  and  the  pressure  on  it 
5G  lbs.,  find  the  pressure  transmitted  to  a  surface  at  E,  the 
area  of  which  is  a  triangle  whose  base  is  |  of  an  inch,  and 
whose  height  is  -^  of  an  inch. 

Ans.  Pressure  on  E==0.42  lbs. 

7.  A  cylindrical  pipe  which  is  filled  with  water  opens 
into  another  pipe  the  diameter  of  which  is  three  times  its 
own  diameter ;  if  a  force  of  20  lbs.  be  applied  to  the  water 
in  the  smaller  pipe,  find  the  force  on  the  open  end  of  the 
larger  pipe  which  is  necessary  to  keep  the  water  at  rest. 

Ans.  180  lbs. 

8.  Kequired  (1)  the  pressure  on  the  sides  of  a  cubical 
vessel  filled  with  water,  and  (2)  the  pressure  on  the  bottom, 
the  side  of  the  vessel  being  a  ft.  (Art.  10). 

Ans.  (1)  125a3  1bs.;  (2)  62.5rtMbs. 

9.  A  cylindrical  vessel  is  filled  with  water ;  the  height  of 
the  vessel  is  a  ft.,  and  tlie  diameter  of  the  base  d  feet. 
Find  (1)  the  pressure  upon  the  side  and  (2)  the  pressure  on 
the  bottom.  Aiis.   (1)  ^\\-r:aH  ;  (2)  15|Tra^. 

♦  lu  tho  first  seven  examples,  the  weight  of  the  liquid  itself  is  not  considered. 


44  EXAMPLES. 

10.  Find  the  height  of  the  vessel  in  Ex.  9  so  that  the 
pressure  on  the  side  may  be  equal  to  the  pressure  on  the 
bottom. 

Ans.  The  height  must  equal  the  radius  of  the  base. 

11.  The  pressure  on  a  square  inch  of  surface  in  a  vessel 
of  mercury  is  1000  grains.  Find  the  pressure  on  a  circular 
surface  of  one-quarter  inch  radius,  placed  9  in.  lower  down, 
mercury  being  13.5  times  as  heavy  as  water. 

Ans.  Pressure  =  0.8886  lbs. 
■  12.  The  water  in  a  canal  lock  rises  to  a  height  of  18  ft, 
against  a  gate  whose  breadth  is  11  ft.     Find  the  total  press- 
ure against  the  gate.  Ans.  Pressure  =  49|-  tons.* 

13.  The  upper  side  of  a  sluice-gate  is  1Q\  ft.  beneath  the 
surface  ;  its  dimensions  are  3  ft.  vertical  by  18  in.  horizon- 
tal.   Find  the  pressure  upon  it. 

A71S.  Pressure  =  1\  tons.* 

14.  A  dyke  to  shut  out  the  sea  is  200  yards  long,  and  is 
built  in  courses  of  masonry  one  foot  high ;  the  water  rises 
against  it  to  a  height  of  6  fathoms.  Find  the  pressure 
against  the  Ist,  18th,  and  36th  courses. 

(  1st  pressure  =  610. •!  tons.* 
Ans.  I  2d  jiressure  ==  318.1  tons. 
(  3d  pressure  =      8.(5  tons. 

15.  Find  the  pressure,  in  pounds,  of  a  cylinder  of  water 
4  inches  in  diameter  and  45  ft.  in  height. 

Ans.  Pressure  =  244.8  lbs. 

16.  A  cubical  vessel,  each  side  of  which  is  10  ft.,  is  filled 
with  water,  and  a  tube  32  ft.  long  is  fitted  to  an  aperture  in 
it,  whose  area  is  one  square  inch.  If  the  tube  be  vertical, 
and  of  the  same  size  as  the  aperture,  and  filled  with  water, 
find  the  pressure  on  the  interior  surface  of  the  vessel,  (1) 
neglecting  the  weight  of  the  water  it  contains,  (2)  when  the 
weight  of  the  water  is  taken  into  account. 

Ans.  (1)  1,200,000  lbs. ;  (2)  1,387,500  lbs. 

•  One  ton  =  2240  lbs. 


EXAMPLES.  45 

17.  Find  the  pressure  on  a  square  inch  at  a  depth  of 
100  ft.  in  a  lake,  (1)  neglecting,  (2)  taking  account  of  the 
atmospheric*  pressure.       Ans.  (1)  43f|  lbs. ;  (2)  58  lbs. 

18.  A  reservoir  of  water  is  200  ft.  above  the  level  of  the 
ground  floor  of  a  house ;  find  the  pressure  of  the  water,  per 
square  inch,  in  a  pipe  at  a  height  of  30  ft.  above  the  ground 
floor,  neglecting  atmospheric  pressure.       Ans.  1^\W  lbs. 

19.  An  equilateral  triangular  area  is  immersed  vertically 
in  water  with  a  side,  one  foot  in  length,  in  the  surface. 
Find  the  pressure  upon  it  in  ounces.  Ans.  125  oz. 

20.  A  hollow  cone,  vertex  upwards,  is  just  filled  with 
liquid.  Find  {\)  the  pressure  on  its  base,  (2)  the  normal 
pressure  on  its  curved  surface,  (3)  the  vertical  pressure  on 
the  curved  surface.  [Let  r  =  the  rndius  of  the  base  and 
]i  =  the  altitude.] 

Ans.  {l)(/pnrVi;  (2)  IgpnrhVr^h^;  (S)  ^gpTrr^h. 

21.  A  vertical  rectangle  has  one  side  in  the  surface  of  a 
liquid.  Divide  it  by  a  horizontal  line  into  two  parts  on 
which  the  pressures  are  equal. 

Atis.  If  h  be  the  vertical  side,  the  depth  of  the  horizontal 

T  ^ 

line  =  --=• 

V2 

22.  A  vertical  triangle,  altitude  h,  has  its  base  horizontal 
and  its  vertex  in  the  surface.  Divide  it  by  a  horizontal  line 
into  two  parts  on  which  the  pressures  are  equal. 

A)is.  The  depth  =  ^-7=' 
V2 

23.  A  smooth  vertical  cylinder  one  foot  in  height  and  one 

foot  in  diameter  is  filled  with  water,  and  closed  by  a  heavy 

piston  weighing  4  lbs.     Find   the  whole  pressure   on   its 

curved  surface.  .        ^ .       1257r  „ 

Ans.  16  -I ^  lbs. 

*  See  Art.  !1,  Cor.  8. 


46  EXAMPLES. 

24.  A  hollow  cylinder,  closed  at  both  ends,  is  just  filled 
with  water  and  held  with  its  axis  horizontal ;  if  the  whole 
pressure  on  its  surface,  inchiding  the  plane  ends,  be  three 
times  the  weight  of  the  fluid,  compare  the  height  and 
diameter  of  the  cylinder.  Ans.  As  1:1. 

25.  The  side  AB  of  a  triangle  ABC  is  in  the  surface  of  a 
fluid,  and  a  point  D  is  taken  in  AC,  such  that  the  pressures 
on  the  triangles  BAD,  BDC,  are  equal.  Find  the  ratio 
AD  :  DC.  Am.  As  1  :  V2  -  1. 

26.  The  diameters  of  the  two  pistons,  p  and  P  (Fig.  4), 
are  2^  in.  and  9  in.,  respectively,  and  the  smaller  is  60  in. 
above  the  larger.  What  force  must  be  applied  to  the 
smaller  piston  that  the  larger  may  exert  a  pressure  of 
1600  lbs.  ?  Ans.  112.8  lbs. 

27.  Compare  the  pressure  on  the  area  of  a  parabola  with 
that  on  its  circumscribing  rectangle,  both  being  immersed 
perpendicularly  to  the  vertex.  Ans.  As  4:5. 

28.  A  cubical  vessel  is  filled  with  two  liquids,  of  given 
densities,  the  volume  of  each  being  the  same.  Find  the 
pressure  on  the  base  and  on  any  side  of  the  vessel. 

Let  rt  be  a  side  of  the  vessel,  p  and  p'  the  densities  of  the 
upper  and  lower  liquids,  p'  being  greater  than  p. 

The  pressure  on  the  base  =  the  weight  of  the  whole 
fluid 

The  pressure  on  the  upper  half  of  any  side 

a^  a        1       „ 
=  ^^2'4  =  8^^^'- 

To  find  the  pressure  on  the  lower  half,  replace  the  upper 
liquid  by  an  equal  weight  of  the  lower  liquid,  which  will 
not  affect  the  pressure  at  any  point  of  the  lower  half.  If  a 
be  the  height  of  this  equal  weight,  we  have 


EXAMPLES. 


4tl 


and  the  depth  of  the  centre  of  gravity  of  the  lower  half 
below  the  upper  surface  of  the  equal  weight 


=  «'  + 


4  ~  4  \    ^  p'f' 


therefore,  the  pressure  on  the  lower  half 


=  (JP 


(^  - 1) 


=  ya^  ip'  +  2p). 

(Besant's  Hydrostatics,  p.  37.) 

29.  A  circle  is  just  immersed  vertically  in  a  fluid.  Find 
on  which   chord,  drawn  from  the  lowest 

point,  the  pressure  is  the  greatest. 

[Let  ADBC  be  the  circle  with  radius  a 
and  BC  the  required  chord,  which  bisect  in 
H,  and  draw  HK  perpendicular  to  AB; 
.*.  etc.]  Ans.  AK  =  ^a. 

30.  A  semicircle  is  immersed  vertically 
in  a  fluid,  with  its  diameter  in  the  upper 
surface;  find  on  which  chord,  parallel  to  the  surface,  the 
pressure  is  the  greatest,  supposing  the  density  of  the  fluid 
to  increase  as  the  depth. 

[Let  LBM  (Fig.  20)  be  the  semicircle,  and  DE  the  chord 
on  which  the  pressure  is  the  gi-eatest,  and  a  the  radius  of 
the  circle.  Then  if  the  density  were  uniform,  the  pressure 
would  vary  as  DG  x  GrF  (Art.  15) ;  but,  since  the  density 
varies  as  the  depth,  the  pressure  varies  as  DG  x  GF^ ; 
•••  etc.]  A71S.  FG  =  aVl 

31.  If  LBM  (Fig.  20)  be  a  parabola,  FB  =  b.  the  latus 
rectum  =  4ft,  and  the  other  conditions  the  same  as  in 
Ex.  30,  find  FG,  the  depth  of  the  chord  of  greatest  pressure 
below  the  upper  surface.  Ans.  FG  =  |i. 


48  .  EXAMPLES. 

32.  The  lighter  of  two  fluids,  whose  densities  are  as  2  :  3, 
rests  on  the  heavier,  to  a  depth  of  4  in.  A  square  is  im- 
mersed in  a  vertical  position,  with  one  side  in  the  upper 
surface.  Determine  the  side  of  the  square  in  order  that  the 
pressures  on  the  portions  in  the  two  fluids  may  be  equal. 

A71K  f  (l  +  V'i0)in. 

33.  Find  the  centre  of  pressure  of  a  semi-parabola,  the 
extreme  ordinate  coinciding  with  the  surface  of  the  fluid. 

[Tjet  LBF  (Fig.  20)  be  the  semi-parabola;  let  BF  =  ^r, 
and  LF  =  b,  and  suppose  0  to  be  the  centre  of  pressure, 
OG  being  parallel  to  LF.]       Am.  FG  =  ia  ;  GO  =  ^^b. 

34.  A  quadrant  of  a  circle  is  just  immersed  vertically  in 
a  liquid,  with  one  edge  in  the  surface,  as  in  Ex.  3,  Art.  16. 
Find  the -centre  of  pressure  when  the  density  varies  as  the 
iiepth. 

Taking  the  edge  in  the  surface  for  the  axis  of  y  and  the 
vertical  edge  for  the  axis  of  x,  we  find 

-  _  32  fl  -  _  1^  « 

35.  The  total  breadth  of  a  water  passage  closed  by  a  pair 
of  flood-gates  is  10  ft.  and  its  depth  is  6  ft.  :  the  hinges  are 
placed  at  one  foot  from  the  top  and  bottom.  Find  the 
pressure  upon  the  lower  hinge  when  the  water  rises  to  the 
top  of  the  gates.  J/^s.  4218|  lb3. 

36.  If  we  suppose  everything  to  be  the  same  as  in  Ex.  2, 
Art.  20,  except  that  the  height  of  the  wall  is  determined  by 
the  condition  that  the  wall  just  sustain  the  pressure  when 
the  water  rises  to  the  top,  what  is  the  height  of  the  wall  ? 

Ans.  6.96  ft. 

37.  A  wall  of  masonry,  a  section  of  which  is  a  rectangle, 
is  10  ft.  high,  3  ft.  thick,  and  each  cubic  foot  weighs  100 
lbs.  Find  the  greatest  height  of  water  it  will  sustain  with- 
out being  overturned.  Ana.  O-v/2 


EXAMPLES.  49 

38."  If  the  height  of  the  wall  be  8  ft.,  its  thickness  6  ft., 
and  each  cubic  foot  weighs  180  lbs.,  find  whether  it  will 
stand  or  fall  when  the  water  is  on  a  level  with  the  top. 

Ans. 

39.  The  depth  AB  of  the  water  in  the  head  bay  (Fig.  13) 
is  7  ft.,  the  depth  DB  of  the  water  in  the  chamber  of  the 
lock  is  4  ft.,  and  the  Avidtli  of  the  lock-chamber  is  7.5  ft.  ; 
find  (1)  the  resnltant  pressure  upon  the  gate  AB,  and  (2) 
the  depth  of  the  point  of  application  of  the  resultant  press- 
ure below  the  surface  of  the  water  in  the  head  bay. 

Ans.  (I)  7734.4  lbs.;  (2)  4.18  ft. 

40.  If  the  vessel  (Fig.  15)  make  140  turns  per  minute, 
find  the  value  of  NM.  Ans.  1.78  in. 

41.  A  liollow  paraboloid  of  revolution,  with  its  axis  ver- 
tical and  vertex  downwards,  is  half  filled  with  liquid.  With 
what  angular  velocity  must  it  be  made  to  rotate  about  its 
axis,  in  order  that  the  liquid  may  just  rise  to  the  rim  of  the 

Ans.  If  2jo  =  latus  rectum,  (J^  =  ^• 

42.  If  the  vessel  in  the  last  example  be  filled  with  liquid, 
find  the  angular  velocity  and  the  time  of  rotation  that  it 
may  just  be  emptied. 

Ans.  If  2»  =  latus  rectum,  (,^  =  ^\   time  =  2;r\  /?. 

P  \  9 

43.  A  hemispherical  bowl  is  filled  with  liquid,  which  is 

made  to  rotate  uniformly  about  the  vertical  radius  of  the 

bowl.     Find  how  much  runs  over.  ,        1  na^iJ^ 

Ans.  -r 

.  ^ 

44.  A  closed  cylindrical  vessel,  height  h  and  radius  a,  is 

just  filled  with  liquid,  and  rotates  uniformly  about  its  ver- 
tical axis.  Find  the  pressures  on  its  tTpper  and  lower  ends, 
and  the  whole  pressure  on  its  curved  surface. 

Ans.  gpna^^,  gp-na^^^^hj,  and  gpTTah\1i-ir^y 


CHAPTER    II. 

EQUILIBRIUM    OF    FLOATING    BODIES.  —  SPECIFIC 
GRAVITY. 

24.  Upward  Pressure,  Buoyant  Effort. — To  find 
the  resultant  pressicre  of  a  liquid  on  the  surface  of  a 
solid  either  wholly  or  partially  immersed. 

Let  ABCD  be  a  solid  floating  in  a  liquid  whose  upper 
surface  is  EF.     Imagine  this  solid  removed,  and  the  space 
it  occupied  filled  wath  the  liquid,  and  suppose  this  liquid  to 
be  solidified.     It  is  clear  that  the  result- 
ant pressure  upon  this  solidified   liquid     E_ F 

will  be  the  same  as  uj)on   the  original      &^^^^^fj 
solid.     But  this  solidified  mass  is  at  rest      ^^^^^Isf^ 
under  the  action  of  its  own  weight  and       Wf~^^^^^ 
the  pressure  of  the  surrounding  liquid ;        ^^^^^^^ 
and,  as  its  own  weight  acts  vertically  p^    jj 

downward  through  its  centre  of  gravity, 
the  resultant  pressure  of  the  surrounding  liquid  must  be 
equal  to  the  weight  of  the  solidified  mass,  and  must  act  ver- 
tically upwards  in  a  line  passing   through   its   centre   of 
gravity. 

The  above  reasoning  is  equally  applicable  to  the  case  of  a 
body  immersed  in  elastic  fluid. 

Therefore,  if  a  solid  be  either  wholly  or  partially  im- 
mersed in  a  fluid,  it  loses  as  much  of  its  weight  as  is 
equal  to  the  weight  of  the  fluid  it  displaces.* 

*  The  discovery  of  this  principle  is  due  to  Archimedes.  (Goodeve,  p.  190 ;  Gal- 
braitb,  p.  49.) 


UPWARD   FKESSURE,    BUOY  AST   EFFORT.  51 

Cor.  1. — If  ii  body  be  supported  entirely  by  a  fluid,  the 
weiglit  of  the  body  must  be  equal  to  the  weight  of  the  fluid 
displaced,  and  the  centres  of  gravit}"  of  the  body  and  of  the 
fluid  displaced  must  lie  in  the  same  vertical  line. 

ScH. — These  conditions  hold  good,  whatever  be  the  nature 
of  the  fluid  in  which  the  body  is  floating.  If  it  be  hetero- 
geneous, the  displaced  fluid  mus|  consist  of  horizontal  strata 
of  the  same  kind  as,  and  continuous  with,  the  horizontal 
strata  of  uniform  density,  in  which  the  particles  of  the  sur- 
rounding fluid  are  necessarily  arranged.  If,  for  instance,  a 
solid  body  float  in  water,  partially  immersed,  its  weight  will 
be  equal  to  the  weight  of  the  water  displaced,  together  with 
the  weight  of  the  air  displaced. 

The  upward  pressure  of  a  fluid  against  a  solid,  and  which 
is  equal  to  the  weight  of  the  displaced  fluid,  is  called  the 
buoyant  effort  of  a  fluid.  The  centre  of  gravity  of  the  dis- 
placed fluid  is  called  tlie  centre  of  buoyancy.  The  buoyant 
effort  exerted  by  a  fluid  acts  vertically  upwards  through  the 
centre  of  buoyancy. 

The  enunciation  and  proof  of  this  proposition  are  due  to  Archi- 
medes, and  it  is  a  remarkable  fact  in  tiie  liistory  of  science,  that  no 
furtlier  progress  was  made  in  Hydrostatics  for  1800  years,  and  until 
the  time  of  Stevinus,  Galileo,  and  Torriceili,  the  clear  idea  of  fluid 
action  thus  expounded  by  Archimedes  remained  barren  of  results. 

An  anecdote  is  told  of  Archimedes,  which  practically  illustrates  the 
accuracy  of  his  conceptions.  Hiero,  king  of  Syracuse,  had  a  certain 
quantity  of  gold  made  into  a  crown,  and  suspecting  that  the  goldsmith 
had  abstracted  some  of  the  goid  and  used  a  portion  of  alloy  of  the  same 
weight  in  its  i)lace,  he  applied  to  Archimedes  to  investigate  whether 
such  was  the  case,  and  to  ascertain  the  nature  of  the  alloy.  It  is  re- 
lated that  while  Archimedes  was  in  his  bath,  reflecting  over  the  diffi- 
cult problem  which  the  king  had  given  him.  he  observed  the  water 
running  over  the  sides  of  the  bath,  and  it  occurred  to  him  that  he  was 
dis[)lacing  a  quantity  of  water  equal  in  volume  to  that  of  his  own  body, 
and  therefore  that  a  quantity  of  pure  gold  equal  in  weight  to  the 
crown  would  displace  less  water  than  the  crown,  the  volume  of  any 
weight  of  alloy  being  greater  than  that  of  an  equal  weight  of  gold. 


52  EqUILIBRIUM  OF  AN  IMMERSED   SOLID. 

He  concluded  at  once  that  he  could  completely  solve  the  king's  prob- 
lem,  by  weighmg  the  crown  in  water.  Overjoyed  with  his  discovery, 
he  ran  directly  into  the  street,  crying  out,  "  Eureka !  Eureka !  " 

The  two  books  of  Archimedes  which  have  come  down  to  us  were  first 
found  in  old  Latin  MS.  by  Nicholas  Tartaglia,  and  edited  by  him  in 
1537.  These  books  contain  the  solutions  of  a  number  of  problems  on 
the  equilibrium  of  paraboloids,  and  various  problems  relating  to  the 
equilibrium  of  portions  of  spherical  bodies. 

The  authenticity  of  these  books  is  confirmed  by  the  fact  that  they 
are  referred  to  by  Strabo,  who  not  only  mentions  their  title,  but  also 
quotes  from  the  first  book. 

25.  Conditions  of  Equilibrium  of  an  Immersed 
Solid. — Let  V  denote  the  volume  and  p  the  density  of  the 
sohd;  v'  the  volume  and  p'  the  density  of  the  displaced  fluid: 
the  weights  of  the  solid  and  of  the  displaced  fluid  will  be 
respectively  gpv  and  gp'v' ;  then,  if  the  solid  rest  in  equilib- 
rium in  the  fluid,  we  shall  have 

If  we  suppose  the  solid  to  be  entirely  immersed,  the  vol- 
umes V  and  v'  will  be  equal,  and  the  densities  p  and  p'  must 
also  be  equal  if  the  solid  remains  in  equilibrium,  having  no 
tendency  either  to  ascend  or  descend. 

But  if  the  weight  of  the  immersed  solid  be  greater  than 
that  of  the  fluid  displaced,  we  shall  have 

f/pv  >  ffp'v; 

and  the  solid  will  be  urged  doionwards  by  a  force  equal  to 
gpv  —  gp'v. 

If,  on  the  contrary,  the  weight  of  the  solid  be  less  than 
that  of  the  fluid,  we  shall  have 

gpv  <  gp'v ; 

and  the  solid  will  be  urged  upwards  by  a  force  equal  to 
gp'v  —  gpv. 


EQUILIBRIUM  OF  AN  IMMERSED   SOLID.  53 

That  is,  the  luhoUy  inimersed  solid  will  descend,  re- 
main at  rest,  or  ascend,  according  as  its  density  is 
greater  than,  equal  to,  or  less  than  the  density  of  the 
fluid. 

In  the  first  case  the  solid  will  descend  to  the  bottom,  and 
press  it  with  a  force  equal  to  the  excess  of  its  weight  above 
that  of  an  equal  bulk  of  fluid. 

In  the  third  case  the  solid  will  rise  to  the  surface,  and  be 
but  partially  immersed,  the  volume  v'  of  the  fluid  displaced 
by  the  solid  having  the  same  weight  as  the  entire  solid. 

[An  egg,  placed  in  a  vessel  of  fresh  water,  sinks  to  the 
bottom  of  the  vessel,  its  mean  density  being  a  little  greater 
than  that  of  the  water.  If,  instead  of  fresh  water,  salt 
water  is  employed,  the  egg  floats  at  the  surface  of  the  liquid, 
which  is  a  little  denser  than  the  egg.  If  fresh  watf.r  is 
carefully  poured  on  the  salt  water,  a  mixture  of  the  two 
liquids  takes  place  where  they  are  in  contact ;  and  if  the 
egg  is  put  in  the  upper  part,  it  will  descend,  and,  after  a 
few  oscillations,  remain  at  rest  in  a  layer  of  liquid  of  which 
it  displaces  a  volume  whose  weight  is  equal  to  its  own.] 

Cor. — From  (1)  we  have 

V  :  v'  '.'.  p'  '.  p; 

therefore,  if  a  homogeneous  solid  float  in  a  fluid,  its 
whole  volume  is  to  the  volume  of  the  displaced  fluid  as 
the  density  of  the  fluid  is  to  the  density  of  the  solid. 

ScH. — When  the  floating  solid  and  fluid  are  both  homo- 
geneous, the  centre  of  gravity  of  the  part  immersed  will 
coincide  with  the  centre  of  buoyancy. 

The  section  of  a  floating  body  formed  by  the  plane  of  the 
surface  of  the  fluid  in  which  the  body  floats  is  called  the 
plane  of  flotation.     The  line  passing  through  the  centre  of 


54  EXAMPLES. 

gravity  of  the  floating  body  and  the  centre  of  buoyancy  is 
called  the  axis  of  flotation. 

The  weight  gpv  of  the  body  acting  downwards,  and  the 
Duoyant  effort  gp'v'  acting  upwards  (Art.  24,  Sch.),  form  a 
couple,  by  which  the  body  rotates  till  the  directions  of  these 
forces  coincide,  i.e.,  till  the  centre  of  gravity  of  the  body 
and  the  centre  of  buoyancy  come  into  the  same  vertical  line. 

EXAMPLE. 

1.  A  piece  of  oak  containing  33  cubic  inches,  floats  in 
water ;  how  much  water  will  it  displace,  the  density  of 
the  oak  being  0.743  times  that  of  water  ? 

Ans.  23.776  cu.  in. 

26.  Depth  of  Flotation.* — Tlie  depth  to  which  a 
body  siiihs  helow  its  plans  of  flotation  is  called  its 
Depth  of  Flotation.  When  the  form  and  weight  of  a 
floating  body  are  known,  its  depth  of  flotation  can  be  calcu- 
lated. 

Denoting  the  volume  and  density  of  the  body  by  v  and  p, 
and  of  the  displaced  fluid  by  v'  and  p,  respectively,  we  have 

[Art.  25,  (1)]. 

gpv  =  gp'v' ; 

.-.    v'=:P-,v,  (1) 

by  which  the  depth  of  flotation  can  be  determined,  when- 
ever v'  can  be  determined  in  terms  of  that  depth. 

EXAMPLES. 

1.  Let  the  solid  be  a  right  cylinder,  whose  axis  a  is  verti- 
cal, and  the  radius  of  whose  b^se  is  r;  let  x  denote  the 
depth  of  flotation.     Then  we  have 


Called  also  depth  of  immersion. 


EXAMPLES. 

V 

= 

TTT^a, 

v' 

= 

■nr^x; 

1 

X 

= 

v' 
-  a 

V 

55 


and 


=  ^,a[froin(l)]. 

2.  Let  the  body  be  a  right  cone,  floating  with  its  apex 
belo'.v  the  surface  of  the  fluid  and  the  axis  a  vertical.  Re- 
\juired  the  depth  of  flotation. 

Since  the  vohmies  of  similar  cones  are  proportional  to  the 
cubes  of  their  heights,  we  have,  x  being  the  required  depth, 

t  —  t 

which  in  (1)  gives, 

sfi  p 

3/P 


X  ^  a\ 

3.  Let  the  body  be  a  sphere  of  radius  a,  floating  in  a  fluid. 
Required  the  depth  of  flotation. 

Here  the  displaced  fluid  has  the  form  of  a  segment  of  a 
sphere ;  hence,  calling  x  the  depth,  we  have,  from  mensura- 
tion, 

v'  =  -x^  {a  —  ^x), 

and  V  =  |-«^; 

r'  _  3x^  (a  —  ^x) 

=  ^,  [from(l)]; 

9 

we  have,  therefore,  to  solve  a  cubic  equation  in  order  to  find 
the  depth  of  flotation  of  the  sphere. 


56  EXA3IPLES. 

4.  Let  the  body  be  a  cylindrical  pontoon,*  with  plane 
ends,  and  having  its  axis  horizontal.     Required  to  find  the 
load  requisite  to  sink  the  pontoon 
to  a  given  depth. 

Let  AD  be  the  intersection  of  tiie 
plane  of  flotation  with  the  end 
which  is  a  right  section.  Put  A  = 
the  area  ADK,  the  plane  surface  of 
immersion,  and  /  =  AB,  the  length 
of  the  cylinder;  and  let   Tr=  the 

required  load  that  will  sink  it  to  the  depth  HK.     Then, 
calling  p'  the  density  of  the  fluid,  we  have 

volume  of  displaced  fluid  =  Al,  (1) 

and  weight  of  displaced  fluid  =  gp'Al 

.-.     W  =  gp'AL  (2) 

A  may  be  found  as  follows :  let  r  =  CK,  and  6  =  angle 
ACK ;  then  we  have,  from  mensuration, 

which  in  (2)  gives, 

W  =  gp'r^ln.-^ +  ^sin2d)h  (4) 

(rhich  is  the  required  load. 

Cor.  1.— If  d  =  165°,  we  have,  from  (4), 

W=igp'rHi^rr  +  l)l  (5) 


*  Pontoons  are  portable  boats,  covered  with  balks,  planks,  etc.,  for  forming 
floating  bridges  over  rivers.  They  are  now  usually  made  of  tin,  in  the  shape  of  a 
cylinder,  with  heiD?«nherical  ends.    <^T»te'8  Mechanical  Philosophy.) 


STABILITY  OF  EQUILIBRIUM.  57 

Cor.  2. — If  the  fluid  be  water,  (5)  becomes 

W  =  ir2  (-V-^  +  1)  I  62.5  (Art.  10,  Cor.  1).         (6) 

5.  Let  the  body  be  a  cone  floating  with  its  base  under 
the  fluid,  and  the  axis  a  vertical.  Find  the  depth  of  flota- 
tion. 3/-       ^ 

Ans.  a  —  a\  /  1  —  -  • 
V  P 

6.  A  man  whose  weight  is  150  lbs.  and  density  1.1,  just 
floats  in  water  by  the  help  of  a  quantity  of  cork.  Find  the 
volume  of  the  cork  in  cubic  feet,  its  density  being  .24,  call- 
ing the  density  of  water  1.  A71S.  ^^^  of  a  cubic  foot. 

27.  Stability  of  Equilibrium If  a  floating  body  is 

in  equilibrium,  the  centres  of  gravity  and  of  buoyancy  are 
in  the  same  vertical  line  (Art.  24,  Cor.  1).  Imagine  the 
body  to  be  shghtly  displaced  from  its  position  of  equilibrium 
by  turning  it  round  through  a  small  angle,  so  that  the  axis 
of  flotation  shall  be  inclined  to  the  vertical.  If  the  body  on 
being  released  return  to  its  original  position,  its  equilibrium 
is  stable  ;  if,  on  the  other  hand,  it  fall  away  from  that  posi- 
tion, its  original  position  is  said  to  be  one  of  unstable  equi- 
librium ;  when  the  body  neither  tends  to  return  to  its 
original  position,  nor  to  deviate  farther  from  it,  the  equilib- 
rium is  said  to  be  one  of  indifference. 

The  investigation  of  this  problem  in  its  utmost  extent 
would  lead  to  very  tedious  and  complex  operations,  which 
would  clearly  be  beyond  the  limits  of  this  treatise  ;  we  shall 
therefore  premise  the  three  following  hypotheses,  in  order 
that  we  may  obtain  comparatively  simple  results : 

1.  The  floating  body  will  be  regarded  as  symmetrical 
with  respect  to  a  vertical  plane  through  its  centre  of  gravity 
when  the  whole  is  at  rest,  so  that  we  need  consider  only  the 
problem  for  the  area  of  a  plane  section  of  the  body. 


58  STABILITY  OF  EQUILIBRIUM. 

2.  The  displacement  will  be  regarded  as  very  small. 

3.  The  vertical  motion  of  the  centre  of  gravity  of  the 
body  will  be  disregarded,  as  indefinitely  small. 

Let  EDF  represent  a  body  which  has  changed  from  its 
upright  to  its  present  inclined  position,  by  turning  tiirough 
a  small  angle ;  let  ABD  repre- 
sent the  immersed  part  of  the 
body  before  displacement,  and 
HKD  that  immersed  after  dis- 
placement, and  G  and  0  the 
centres     of    gravity    and    of 
buoyancy  before  displacement. 
While  the  body  moves  from  its 
upright  to  its  inclined  position, 
its  centre  of  buoyancy  moves     ="^-=^ ^^^^"^^3=^^^--"? 
from  0  to  0',  which  latter  is  '"'a" 

in  the  half  of  the  body  most 

immersed,  and  the  wedge-shaped  part  ACH  passes  up  out 
of  the  water,,  drawing  tlie  wedge-shaped  part  BCK  down 
into  it.     Let  the  vertical  line  through  0'  meet  GO  in  M. 

Now  since  the  buoyant  effort  is  equal  to  the  weight  of  the 
whole  solid  (Art.  24,  Sch.),  the  magnitude  of  the  part  im- 
mersed will  be  unaltered  ;  therefore  ABD  =  HKD,  and 
ACH  z=  BCK  ;  also,  the  buoyant  effort  P,  acting  at  0' 
vertically  upwards,  and  the  weight  P  of  the  solid,  acting 
at  G  vertically  downwards,  form  a  couple  which  tends  to 
restore  the  body  to  its  original  position  when  M  is  above  G; 
and,  on  the  contrary,  it  tends  to  incline  the  body  farther 
from  its  original  position  when  M  is  heloio  G.  Hence,  the 
stability  of  a  floating  body,  a  ship,  for  instance,  depends 
upon  the  position  of  the  point  M,  where  the  vertical  line 
through  the  centre  of  buoyancy,  in  the  inclined  position  of 
the  body,  cuts  the  line  connecting  the  centre  of  gravity  and 
centre  of  buoyancy  in  the  upright  position  of  the  body. 

The  position  of  the  point  M  will  in  general  depend  on  the 
extent  of  displacement.     K  the  displacement  be  very  small. 


STABILITY  OF  EQUILIBRIUM.  59 

i.  e.,  if  the  angle  between  GO  and  the  vertical  be  very  small, 
the  point  M  is  called  the  metacentrc,  and  the  question  of 
stability  is  now  reduced  to  the  determination  of  this  point. 
A  ship,  or  any  other  body,  floats  with  stability  when  its 
metacentre  lies  above  its  centre  of  gravity,  and  without  sta- 
bility when  it  lies  below  it;  it  is  in  indifferent  equilibrium 
when  these  two  points  coincide.  Hence  the  danger  of 
taking  the  whole  cargo  out  of  a  ship  without  putting  in 
ballast  at  the  same  time,  or  of  putting  the  heaviest  part  of 
a  ship's  cargo  in  the  top  of  the  vessel  and  the  lightest  in 
the  bottom,  or  the  risk  of  upsetting  when  several  peoj)le 
stand  up  at  once  in  a  small  boat. 

One  of  the  most  important  problems  in  naval  architecture 
is  to  secure  the  ascendancy,  under  all  circumstances,  of  the 
metacentre  above  the  centre  of  gravity.  This  is  done  by  a 
proper  form  of  the  midship  sections,  so  as  to  raise  the  meta- 
centre as  much  as  possible,  and  by  ballasting,  so  as  to  lower 
the  centre  of  gravity.* 

The  horizontal  distance  MN,  of  the  metacentre  M,  from 
the  centre  of  gravity  G  of  the  body,  is  the  arm  of  the  couple 
whose  forces  are  P  and  P,  the  weight  of  the  body  and  the 
buoyant  effort;  and  the  moment  of  this  couple,  which 
measures  the  stability  of  the  body,  isP-MN.  Let  GM  =  c, 
and  the  angle  OMO',  through  which  the  body  rolls,  =  Q, 
and  denote  the  measure  of  the  stability  by  8\  then  we  have 

8=.  P-MN  =  Pcsin  0;  (1) 

therefore,  the  stahility  of  a  hody,  in  general,  varies  as 
its  iceight,  as  the  distance  of  its  metacentre  from  its 
centre  of  gravity,  and  as  the  angle  of  inclination ; 
and  hence,  in  the  same  hody,  for  a  given  inclination, 
it  depends  only  upon  the  distance  of  its  metacentre 
from  its  centre  of  gravity. 

*  Besant's  Hydrostatioe,  p-  58. 


60  POSITION  OF  THE  METACENTRE. 

28.  The  Position  of  the  Metaceiitre;  the  Measure 
of  the  Stability.— Since  the  stability  of  a  body  depends 
principally  upon  the  distance  of  the  metacentre  from  the 
centre  of  gravity  of  the  body,  it  becomes  important  to  de- 
termine the  position  of  tlie  metacentre. 

Let  A  =  the  cross-section  ABD  =  HKD  (Fig.  23)  of 
the  immersed  part  of  the  body  (Art.  27).  and  A^  =  the 
cross-section  ACH  =  BCK  ;  let  (/  and  (/'  be  the  centres  of 
gravity  of  ACH  and  BCK  ;  let  a  =  the  horizontal  distance 
RL,  between  these  centres  of  gravity,  and  s  =  the  horizon- 
tal distance  between  0  and  0',  the  centres  of  buoyancy. 
Then,  taking  moments  round  G,  we  have, 

HKD  X  MX  -  ACH  x  EN  =  ABD  x  NT  +  BCK  x  NL ; 

or,  A  (MN  -  NT)  =  A  ^  (RN  +  NL) ; 

.*.    As  =  A^a'y 

or,  s  =  -r"  a; 

A 


and  OM  = 


00'  A^a 


sin  0        A  sin  0' 


which  is  the  height  of  the  metacenti'e  above  the  centre 
of  buoyancy . 

Let  GO  =  e;  then 

.  =  GM  =  .  +  -A5_,  (1) 

ivhich  gives  the  height  of  the  vnetacentre  above  the  cen- 
tre of  gravity. 

Substituting  this  value  of  c  in  (1)  of  Art.  27,  we  get 

8=P  (^  +  e  sin  0),  (2) 

which  is  the  measure  of  the  stability. 


MEASURE   OF  STABILITY.  61 

If  the  point  0  were  below  G..  e  would  be  negatire  and  (2) 
would  be 

S=  P  {^^  -  e  sin  0).  (3) 

Hence,  in  general,  we  have 

S=p{^±e^\ne),  <4) 

the  upper  or  lower  sign  being  used  according  as  the  centre 
of  buoyancy  is  above  or  below  the  centre  of  gravity. 

Cor.  1. — If  the  displacement  be  small,  the  cross-sections 
ACH  and  BCK  can  be  treated  as  isosceles  triangles,  and 
sin  d  :=  6.  Denoting  the  width  AB  =  HK  of  the  body  at 
che  plane  of  flotation  by  h,  we  have 

A 1  =  \b'^e,         and        EL  =  a  =  ^b, 

which  in  (4)  gives 

^  =  ^(i£±^)''-  <») 

Cor.  2. — When  the  centi-e  of  buoyancy  is  above  the  cen- 
tre of  gravity  of  the  body,  tbe  stability  is  positive,  as  also 
in  the  case  when  the  centre  of  buoyancy  is  below  the  centre 

of  gravity  while  e  is  less  than  ;-^^;  in  this  case  tbe  equi- 
librium is  that  of  stability. 

It  e  is  greater  than  ——j ,  and  the  centre  of  buoyancy  is 

below  the  centre  of  gravity  of  the  hody,  the  stability  is  neg- 
ative, or  the  equilibrium  is  tbat  of  indability. 

If  e  is  negative  and  equal  to  fT-r?  the  stability  is  zero, 

and  the  equilibrium  is  that  of  indifference. 

That  is,  the  centre  of  buoyancy  may  be  heloiv  the  centre 
of  gravity  and  yet  the  stability  be  positive,  so  long  as  e  does 


d2  examples. 

not  exceed  ^-r-i,  which   term  is  always  the  distance  be- 
tween  the  metacentre  and  the  centre  of  buoyancy. 

If  the  centre  of  gravity  of  the  body  coincides  with  the 
centre  of  buoyancy,  we  have  e  =  0,  and  (5)  becomes 

s=P~e.  (6) 

Hence,  generally,  the  stctbility  is  positive,  negative,  or 
zero,  according  as  the  metacentre  is  above,  helow,  or 
coincident  with  t?ie  centre  of  gravity  of  the  floating 
body. 

A  vertical  line  O'M  through  the  centre  of  buoyancy  is 
called  a  line  of  snpport. 

Cor,  3. — From  the  above  results  we  see  that  the  stability 
of  a  body  is  greater  the  broader  it  is  and  the  lower  its  centre 
of  gravity  is.  (See  Weisbach's  Mechs.,  Vol.  I.,  p.  760;  also 
Bland's  Hydrostatics,  p.  120.) 

EXAMPLES. 

1.  Determine  the  stability  of  a  homogeneous  rectangular 
parallelopiped  floating  in  a  fluid. 

Let  HK  be  the  line  of  flotation  of  ^"""^^^^^ 

a  vertical  section  passing  through  the  /     /     /'^ 

centre   of  gravity   G ;    let    J  =  the  HJ~-~^       L 

bi-eadth  EF  of  the  section  of  the  par-  fC-^     J^^^h^^^i 

allelopiped,  h  =  the  height  EC,  and  ^S/      /     /^^^g 

y  =  the   depth   of  immersion   AC.  ^&j^>,J^^^^^ 

Then  we  have  ^S^^^^^^^^ 
A  =  by,  and  e  =  —iih—y),  '"'a-^*  ~ 

e  being  negative  since  the  centre  of  buoyancy  is  below  the 
centre  of  gravity.     Substituting  in  (5),  we  have 


EXAMPLES.  63 

Let  the  density  of  the  material  of  the  parallelopiped  be  p 
times  that  of  the  fluid  ;  then  (Art.  25,  Cor.), 


which  in  (1)  gives 


8  = 


p  :  1  ::  y  :  h  ; 

•'■   y  =  hp, 


Li2Fp-2^^-P\ 


Pe,  (2) 


which  is  the  measure  of  the  stability  required. 

Cor.  1. — To  determine  the  limits  of  stability  depending 
upon  the  dimensions  and  density  of  the  solid,  let  /S  =  0, 
and  (2)  becomes 

b^-  6¥p{i  -p)  =  0;  '  (31 

or,  ^  =  ^^(TZT^, 

It  p  =  ^,  we  have 

I  =  iVe  =  1.225, 

and  hence  in  this  case  the  parallelopiped  floats  in  stable 
indifferent,  or  unstable  equilibrium,  according  as  the  breadth 
is  >,  =,  or  <  1.225  times  the  height. 

CoE.  2. — Solving  (3)  for  p,  we  get 


1       1     A        2J2 
^  =  2±2\/  3^. 


h 

which  is  real  when  ^  is  =  or  <  ^\/^  ',   i-  &-,  when  the 
ratio  of  the  breadth  of  the  solid  to  the  height  is  equal  to,  or 


64  EXAMPLES. 

less  than  |a/6,  two  values  may  be  assigned  to  the  density 
of  the  solid  which  will  cause  it  to  float  in  indifferent  equi- 
librium. 

K,  for  instance,  h  =  //,  we  have 

P  =  i  It:  iVr^^l  =  0.78868  or  0.21132. 

Cor.  3. — When  j  >  ^V6,  the  value  of  p  is  imaginary, 

L  e.,  if  the  ratio  of  the  breadth  of  the  solid  to  the  height  is 
greater  than  ^V6,  no  value  can  be  given  to  the  density 
which  will  cause  the  stability  to  vanish.  In  this  case  the 
solid,  placed  with  EF  horizontal,  must  in  all  cases  continue 
to  float  permanently  in  that  position,  whatever  may  be  the 
density,  providing  it  is  always  less  than  that  of  the  fluid. 

Cor.  4.^^-The  term  --—  in  (1),  or  ■  ,^,     in  (2),  is  the  dis- 
12y       ^  '        VZIip       ^  ' 

tance  between  the  centre  of  buoyancy  and  the  metacentre. 

2.  Determine  the  a,ngle  of  incliiiaticn  d.  In  order  ihac  tne 
parallelopiped    EFDC   may 
be  in  a  position  of  indiffer- 
ent equilibrium. 

Let  h  =  the  breadth  EF 
of  the  section  of  the  paral- 
lelopiped;,  y  =  the  depth 
of  immersion  AC  =  BD, 
and  d  =  angle  AOH.    Then     1^ ^  __^_ 

A  =  ABDC  =  HKDC        ^Th:'^'^'^^=^-^^-=^ 

=  hy,  (1)  ^''•^' 

A,  =  AOH  =  BOK. 

But    AO  =  OB  =  \h,     and     AH  =  BK  =  ^b  tan  0; 
therefore 

A^  =  |J2tan  6.  (2) 


EXAMPLES.  65 

Let  g  and  g'  be  the  centres  of  gravity  of  the  triangles 
A.OH  and  BOK ;  draw  Mr/  parallel  to  AH,  and  gU  and  MQ 
perpendicular  to  HO.     Then 

Mg  =  ^b  tan  d,        and      OM  =  ^b. 

Therefore,  the  liorizontal  distance  of  the  centre  of  gravity 
g  from  the  centre  0 

=  OR  =  OM  cos  6  +  Kg  sin  6 

=  ^b  cos  6  -\-  jrb  tan  6  sin  9 ; 

and  therefore,  for  a  =:  RL  =  2 OR,  we  have 

a  =  ^b  cos  0  -]-  ^b  tan  0  sin  9.  (3) 

Substituting  (1),  (2),  and  (3),  in  (3)  of  Art.  28,  and  put- 
ting ^S'  =  0  for  indifferent  equilibrium,  we  get 

■|52  tan  9  (f  5  cos  0  +  ^5  tan  9  sin  9) 


by 

-  "  J 

L(2  -t-  tan2  9)^  —  2Uy]  sin  0  =  0. 

.-.    sin  0  =  0, 
1 

(4) 

tan  0  =  W2Uy  —  2b\ 

(5) 

and 


The  angle  9  =  0,  in  (4),  is  applicable  to  the  body  when 
in  an  upright  position,  and  that  given  in  (5)  is  applicable 
to  the  body  when  floating  in  an  inclined  position,  and  is 
possible  only  when  b  is  =  or  <  2VSey. 

Cor. — Let  h  =  the  height  EC,  and  p  =  the  density  of 
the  body,  the  density  of  the  fluid  being  unity,  then  we  have 

y  =  Jip,        and        e  =  -  (1  —  p), 
which  in  (5)  gives 

tan  9  =  ^Vl2h^  (1  —  p)  P  ^^^^-  (6) 


66  SPECIFIC   GRAVITY. 

Hence,  when  j  <  -v/6p  (1  —  p),  the  parallelopiped  will 

float  in  an  inclined  position  in  indifferent  equilibrium,  the 
inclination  being  given  by  (6). 


When  y  >  VSp  (1  —  p),  the  value  of  tan  0  is  imaginary, 

i.  e.,  if  the  ratio  of  the  breadth  to  the  height  is  greater  than 
'^%p  (1  —  p),  no  value  can  be  found  for  the  inclination 
which  will  cause  the  stability  to  vanish.  (Compare  with 
last  example.) 

3.  If  the  breadth  of  the  parallelopiped  is  equal  to  its 
height,  and  if  p  =  ^,  find  the  inclination  0,  that  the  paral- 
lelopiped may  float  in  indifferent  equilibrium. 

Ans.  6  =  45° 

29.  Specific  Gravity.  —  Tfie  specific  gravity  of  a 
body  is  the  ratio  of  its  weight  to  the  weight  of  an  equal 
volume  of  some  other  body  taken  as  the  standard  of 
comparison. 

The  density  of  a  body  has  been  defined  (Anal.  Mechs., 
Art.  11),  to  be  the  ratio  of  the  mass  of  the  body  to  the  mass 
of  an  equal  volume  of  some  other  body  taken  as  the  stand- 
ard ;  and  since  the  weights  of  bodies  are  proportional  to 
their  masses,  it  follows  that  the  ratio  of  the  weights  of  two 
bodies  is  equal  to  the  ratio  of  their  masses.  Hence,  the 
measure  of  the  specific  gravity  of  a  body  is  the  same  as  that 
of  its  density,  provided  that  both  be  referred  to  the  same 
standard  substance. 

Thus,  let  S,  W,  V,  and  p  be  the  specific  gravity,  weight, 
volume,  and  density,  respectively,  of  one  body,  and  S^,  PTj, 
Fj,  and  pj  the  same  of  another  body ;  then  we  have 

J'  _  irZ^  _  _pI_  n) 

w;  -  gp^v,  -  p,V,'  ^^ 


THE  STANDARD   TEMPERATURE.  67 

and  making  the  volumes  equal,  we  have 

that  is,  the  ratio  of  the  specific  gravities  of  two  bodies 
is  equal  to  that  of  their  densities. 

Now  suppose  the  body  whose  weight  is  W^  to  be  assumed 
as  the  standard  for  specific  gravity;  then  will  S^  be  unity, 
and  (2)  will  become 

'?=-J-  =  -  (3) 

Also,  if  the  same  body  be  assumed  as  the  standard  of 
density,  p^  will  be  unity,  and  (3)  will  become 


W 


Hence,  the  measure  of  the  specific  gravity  of  a  hd&y 
is  the  same  as  that  of  its  density,  i.  e.,  the  numbers  S 
and  p  are  identical,  when  both  specific  gravity  and 
density  are  referred  to  the  same  substance  as  a 
standard. 

30.  The  Standard  Temperature. — The  standard  sub- 
stance to  which  specific  gravity  and  density  are  referred  is 
not  necessarily  the  same,  and  therefore  8  and  p  will  in  gen- 
eral be  different  numbers.  In  practice,  it  is  usual  to  adopt 
water  as  the  standard  in  determining  the  specific  gravities 
of  solids  and  incompressible  fluids ;  and  for  the  purpose  of 
rendering  the  comparison  more  exact,  the  water  must  first 
be  deprived  by  distillation  of  any  impurities  which  it  may 
contain. 

The  dimensions  of  all  bodies  being  more  or  less  changed 
by  changes  of  temperature,  it  becomes  necessary  to  adopt  a 
standard  temperature  at  which  experiments  for  determining 


68  THE  STANDARD    TEMPERATURE. 

specific  gravities  must  be  performed.  The  Euglisli  *  usually 
take  for  this  purpose  the  temperature  of  60°  Fahrenheit,  it 
being  easily  obtained  at  all  times,  and  the  tables  of  specific 
gravities  are  usually  given  with  reference  to  distilled  water 
at  this  temperature  as  the  standard.  When  the  experiment 
cannot  be  performed  at  the  standard  tempei-ature,  the  result 
obtained  must  be  reduced  to  what  it  would  be  at  this  tem- 
perature, *.  e.,  the  apparent  specific  gravity,  as  obtained  by 
means  of  water  when  not  at  the  standard  temperature,  must 
be  reduced  to  what  it  would  have  been  if  the  water  had 
been  at  the  standard  temperature. 

Thus,  let  p  be  the  density  of  any  solid,  S^  its  apparent 
specific  gravity  as  obtained  by  water  when  not  at  the  stand- 
ard temperature,  and  pj  the  corresponding  density  of  the 
water  ;  and  let  She  the  true  specific  gravity-  of  the  body  as 
determined  by  water  at  a  standard  temperature,  the  corre- 
sponding density  of  the  water  being  pg.  Then,  from  (3)  of 
Art.  29,  we  have 

S.=-^,        and        S==-; 
*        Pi  Pi 

'^i        Pz 
Calling  the  density  of  the  standard  temperature  unity, 
(1)  becomes 

S=S,p,.  (2) 

That  is,  the  specific  gravity  of  a  body  as  determined 
at  the  standard  temperature  of  the  water  is  equal  to 
its  specific  gravity  determined  at  any  other  tempeirt- 
ture,  multiplied  hy  the  density  of  the  water  at  this 
temperature,  the  density  of  the  water  at  the  standard 
temperature  being  regarded  as  unity. 

ScH. — In  the  cases  that  occur  most  frequently  in  prac- 
tice, such  nicety  is  unnecessary,  and  the  experiment  may  be 

*  The  French  nsnally  take  the  temperature  at  which  water  has  its  maximum  ol 
density,  which  is  39''.4  P. 


METHODS    OF  FINDING    SPECIFIC   GRAVITY.  69 

performed  with  water  at  any  temperature;  but  tlie  temper- 
ature must  be  noted  and  a  correction  applied  for  it  which 
depends  upon  the  density  of  water  at  the  experimental 
temperature.* 

The  weight  of  a  cubic  foot  of  distilled  water  at  the  stand- 
ard temperature  is  1000  ozs.  ^  62|^  lbs. ;  hence  we  tind  the 
weight  of  a  cubic  foot  of  any  substance  in  ounces  or  pounds 
by  multiplying  its  specific  gravity  by  1000  or  62^. 

It  appears,  therefore,  that  by  means  of  the  specific  gravi- 
ties of  homogeneous  bodies,  their  weights  may  be  deter- 
mined without  actually  weighing  them,  provided  their 
volumes  are  known  ;  and  conversely,  however  irregular  the 
shape  of  bodies  may  be,  if  their  weights  and  specific  gravi- 
ties are  known,  their  volumes  may  be  determined,  viz.,  by 
dividing  the  weight  by  the  specific  gravity. 

The  specific  gravities  of  gases  and  vapors  are  usually 
determined  by  referring  them  to  atmospheric  air  at  the  same 
temperature  and  under  the  same  pressure  as  the  gases  them- 
selves. 

31.    Methods   of  Finding   Specific   Gravity. — The 

law  of  the  buoyant  effort,  or  upward  pressure,  of  water  can 
be  made  use  of  •to  determine  the  specific  gravities  of  bodies; 
for,  if  a  body  be  immersed  in  a  fluid,  it  loses  as  much  of  its 
weight  as  is  equal  to  the  weight  of  the  fluid  it  displaces 
(Art.  24) ;  i.  e.,  if  it  be  wholly  immersed,  its  loss  of  weight 
is  equal  to  the  weight  of  its  volume  of  the  fluid. 

Thus,  if  a  sphere  of  lead,  whose  weight  is  11  lbs.,  were 
found  to  weigh  but  10  lbs.  when  immersed  in  water,  we 
should  conclude  that  the  weight  of  an  equal  volume  of 
water  would  be  one  pound,  and  therefore  that  the  lead 
weighed  11  times  as  much  as  its  volume  of  water,  and  hence 
that  the  specific  gravity  of  lead  was  11 ;  and  so  for  any  other 
substance. 

•  Eenwick's  Mechs.,  p.  334. 


70  METHODS   OF  FINDISG   SPECIFIC   GRAVITY. 

(1)  To  find  the  specific  gravity  of  a  solid  heavier 
than  water. 

Let  w  =  the  weight  of  the  solid  in  air,  w'  =  its  weight 
in  water,  and  *S'  =  the  specific  gravity  of  the  solid,  that  of 
water  being  1 ;  then  w  —  w'  is  the  weight  lost  by  the  solid, 
which  is  also  the  Aveight  of  the  water  displaced  by  the  solid 
(Art.  24) ;  therefore  w  and  w  —  to'  are  the  weights  of  equal 
volumes  of  the  solid  and  water.     Hence  we  have 

S  =  ^^,.  (1) 

Hence,  to  find  the  specific  gravity  of  a  solid  heavier  than 
water,  we  have  the  following  rule:  Divide  its  weight  hy 
its  loss  of  weight  in  ivater. 

(2)  To  find  the  specific  gravity  of  a  solid  lighter 
than  water. 

Since  the  solid  is  lighter  than  water,  it  will  not  descend 
in  the  water  by  its  own  weight ;  it  must  therefore  be  at- 
tached to  a  heavy  body  of  sufficient  size  and  weight  to  make 
the  two  together  sink  in  the  water. 

Let  w  =  the  weight  of  the  solid  in  air, 

X  =  the  weight  in  air  of  the  heavy  body 

attached  to  it, 
x'  =  the  weight  in  the  water  of  the  heavy 

body, 
w'  =  the  weight  in  the  water  of  the  two 
together. 

Then  w  +  x—io'  =  the  weight  of  water  displaced  by  the 
two  together. 
x—x'  =  the  weight  of  water  displaced  by  the 
heavy  body. 

Hence,  tc-^x'—w'  =  the  weight  of  water  displaced  by  the 
solid,  and  therefore 


EXAMPLES.  71 

8  =  — ? — ;■  m 

to  -\-  X  —  w  ' 

Hence,  add  the  difference  between  the  weights  of  the 
heavy  body  and  the  two  together  in  the  water  to  the 
weight  of  the  solid  in  air,  and  divide  the  weight  of  the 
solid  in  air  by  this  sum. 

(3)   To  find  the  specific  gravity  of  a  liquid. 

Take  a  solid  which  is  specifically  heavier  than  either  the 
liquid  or  water,  and  let  it  be  weighed  in  both  ;  then  the  loss 
of  weight  in  the  two  cases  will  be  the  respective  weights  of 
equal  volumes  of  the  liquid  and  of  water ;  therefore,  tlie 
loss  of  weight  in  the  liquid,  divided  by  the  loss  of 
weight  in  the  water,  will  give  the  specific  gravity  of 
the  liquid. 

Let  w  =  the  weight  of  the  solid  in  air,  w'  =  its  weight 
in  the  liquid  whose  specific  gravity  is  to  be  determined,  and 
n\  =  its  weight  in  water;  then  w  —  w'  and  w  —  w^  are 
the  respective  weights  of  equal  volumes  of  the  liquid  and  of 
water;  therefore 

S=''^^^-  '        (3) 

to  —  Wi  ^ 

Otherwise  thus :  Let  to  =  the  weight  of  an  empty  flask, 
w'  =  its  weight  when  filled  with  the  liquid,  and  tv^  =  its 
weight  when  filled  with  water ;  then  to'  —  tv  and  tv^  —  w 
are  the  respective  weights  of  equal  volumes  of  the  liquid  and 
of  water ;  therefore 

S  =  ^^^-  (4) 

Wj —  tv  ^  ' 

BXAMPLES. 

1.  A  cubical  iceberg  is  100  ft.  above  the  level  of  the  sea, 
its  sides  being  vertical.  Given  the  specific  gravity  of  sea- 
water  =  1.0263,  and  of  ice  =  0.9214  at  the  temperature  of 
32°,  to  find  its  dimensions. 


72  SPECIFIC   GRAVITY  OF  BROKEN  SOLID. 

Let  X  =  the  length  of  one  side, 

X  —  100  =  the  length  of  the  piece  under  water; 
then  we  have  (Art.  25,  Cor.), 

a-3  :  a-3  —  lOOa-2  ::  1.0263  :  0.9214; 
.♦.    a;  :  100  ::  1.0263  :  0.1049; 
.-.    X  =  978.3  ft., 
and  x^  =  936,302,451.687  cu.  ft. 

2.  A  piece  of  limestone,  whose  weight  is  256.34  lbs., 
weighs  in  water  159.13  lbs.     Find  its  specific  gravity. 

Ans.  2.637. 

3.  Find  the  specific  gravity  of  a  piece  of  cork  whose 
weight  is  20  grains.  To  sink  it,  we  attach  a  brass  weight 
which,  when  immersed  in  the  water,  weighs  87.22  grains; 
the  weight  of  the  compound  body  when  immersed  is  23.89 
grains.  Ans.  0.24. 

4.  A  solid  weighing  25  lbs.,  weighs  16  lbs.  in  a  liquid  A, 
and  18  lbs.  in  a  liquid  B.  Compare  the  specific  gravities  of 
A  and  B.  Ans.  9  :  7. 

32.  Specific  Gravity  of  a  Solid  broken  into  Frag- 
ments.— Put  the  broken  pieces  into  a  flask,  fill  the  flask 
with  water,  and  let  its  weight  be  then  w" ;  let  w  be  the 
weight  of  the  solid  in  air,  and  tv'  the  weight  of  the  flask 
when  filled  with  water.     Then 

w"—zv'=we\ght  of  solid  pieces— wt.  of  water  they  displace 

^=20  —  weight  of  water  displaced  ; 

therefore    w  +  w'  —  w"  =  weight  of  water  displaced ; 

A      S=---^' .•  (1) 

w  -f-  w  —  w 


SPECIFIC   GRAVITY  OF  A   MIXTURE.  73 

33.  Specific  Grravity  of  Air.  —  Take  a  large  flask 
which  can  be  completely  closed  by  a  stop-cock,  and  weigh  it 
when  filled  with  air ;  withdraw  the  air  by  means  of  an  aii'- 
pump  and  weigh  the  flask  again ;  finally,  fill  the  flask  with 
water  and  weigh  again.  This  last  weight  minus  the  second 
will  give  the  weight  of  the  water  that  filled  the  flask,  and 
the  first  weight  minus  the  second  will  give  the  weight  of  an 
equal  volume  of  air;  divide  the  weight  of  the  air  by  that  of 
the  water ;  the  result  will  be  the  specific  gravity  of  air  as 
compared  with  that  of  water. 

Let  lu  =  the  weight  of  the  exhausted  flask;  w',  w"  its 
weights  when  filled  with  air  and  water ;  then 

zv'  —  w  =  weight  of  the  air  contained  by  the  flask, 
w"  —  w  =  weight  of  the  water  contained  by  the  flask ; 

therefore,  S  =  — n (1) 

w  —  w  ' 

ScH. — In  the  same  manner  the  specific  gravity  of  any  gas 
can  be  obtained.  The  specific  gravity  of  water  at  20°.5  is 
about  768  times  that  of  air  at  0°  under  the  pressure  of  29.9 
inches  of  mercury  at  0°. 

The  atmosphere  in  which  these  operations  must  be  per- 
formed varies  at  different  times,  even  during  the  same  day, 
in  respect  to  temperature,  the  weight  of  its  column  which 
presses  upon  the  earth,  and  the  quantity  of  moisture  it  con- 
tains. On  these  accounts,  corrections  must  be  made  before 
the  specific  gravity  of  air,  or  that  of  any  gas  exposed  to  its 
pressure,  can  be  accurately  determined.  The  discussion  of 
the  principles  according  to  which  these  corrections  are 
made,  is  given  in  Chap.  III. 

34.  Specific  Gravity  of  a  Mixture.— (1)  TV7ien  the 
volumes  and  specific  gravities  of  the  coinpotients  are 
given. 

Let  V,  v',  v",  etc.,  be  the  volumes  of  the  bodies  of  which 


74  SPECIFIC   GRA  VITY  OF  A  MIXTURE. 

the  specific  gravities  are  s,  s',  s",  etc.  Then,  since  the 
weight  of  the  volume  v  is  62.6sv  (Art.  30,  Sch.),  and  so  for 
the  others,  the  weight  of  the  mixture  is 

62.5  (sv  +  s'v'  +  s"v"  +  etc.)  =  62.5  2  (sv) ; 

and  the  volume  of  the  mixture  is 

V  -}-  v'  +  v"  4-  etc.  =  S  (y) ; 

and  therefore,  if  S  be  the  specific  gravity  of  the  mixture, 
we  have 

62.5iS'  2  (/•)  =  62.5  X  (sv) ; 

^  {v)  ^  ' 

If  by  any  chemical  action  the  volume  becomes  F  instead 
of  2  (v),  the  specific  gravity  will  be 

(2)    WTieii  the  weights  and  specific  gravities  of  the 
components  are  given. 

Let  w,  w',  w",  etc.,  be  the  weights  of  the  bodies,  and  s, 

s',  s",  etc.,  their  specific  gravities.     Then,  as  before,  since 

to  =  62.5sv,  and  so  for  the  others,  the  volumes  are  respect- 

w  w' 

ively  ^-^,  m'fT'^  ^^^'>  ^^^  ^^^  whole  volume  is 

bii.oS       \}/i.DS 

IV  w'      ,     i.     _     1     V  ('U^\ 

62.5s  '^  '62.5s'  +  ^^^'  "  63.5  "^  \7/  ' 

and  the  whole  weight  is 

tv  -}-  w'  +  etc.  ^=  S  (w)  ; 

and  therefore,  if  S  be  the  specific  gravity  of  the  mixture, 
we  have 


WEIGHTS   OF  THE   COMPONENTS   OF  A   MIXTURE.       75 


''^•»^6^5^(7)  =  ^W^ 


«=^1-  (3) 


(?) 


Rem. — Instead  of  taking  1  lb.  as  the  unit  of  weight,  as  we 
have  heretofore  done,  it  is  sometimes  more  convenient  to 
take  the  weight  of  a  unit  of  volume  of  the  standard  sub- 
stance as  the  unit  of  weight ;  thus,  in  the  present  Article, 
we  might  have  made  ^i\  lbs.  the  unit  of  weight,  and  found 
the  weights  of  the  substances  in  terms  of  that  unit. 

35.  The  Weights  of  the  Components  of  a  Mechan- 
ical Mixture.  —  MVieu  the  specific  gravities  of  the  mix- 
ture and  its  components,  and  also  the  weight  of  the 
mixture  are  given,  to  find  the  weights  of  the  compo- 
nents. 

Let  20,  ic',  w"  be  the  weights  of  the  mixture  and  its  com- 
])onents  respectively,  s,  s',  s"  their  respective  specific  gravi- 
ties ;  and  v,  v',  v"  their  volumes.     Then  we  have 

w  =  w'  -\-  w",  (1) 

and  also  v  =z  v'  +  v" ; 

w       w       w" 
and,  therefore,  —  =  -^  +  -jr  (Art.  34,  Rem.).      (2) 

s        s        s 

Combining  (1)  and  (2),  we  obtain, 

^'  =  ^C--7')^{7-7') 

{S"-S)S'  fo^ 


W 


76  THE  HYDROSTATIC  BALANCE. 


=  h rjT-  w.  (4) 

(s  —  s  )  s 

EXAMPLES. 

1.  If  with  78  gallons  of  spirit,  specific  gravity  0.92, 
22  gallons  of  water  be  mixed,  what  is  the  specific  gravity  of 
the  mixture?  ^ns.  0.9376. 

2.  British  standard  gold  contains  11  parts  by  weight  of 
pure  gold,  and  1  part  of  copper.  Required  its  specific 
gravity.  Ans.  17.647. 

8.  An  iron  vessel  completely  filled  with  mercury  weighed 
500  lbs.,  and  lost,  when  weighed  in  water,  40  lbs.  If  the 
specific  gravity  of  the  cast  iron  is  7.2  and  that  of  the  mer- 
cury is  13.6,  find  (1)  the  weight  of  the  empty  vessel,  and 
(2)  that  of  the  mercury  contained  in  it. 

Ans.   (1)  49.5  lbs. ;  (2)  450.5  lbs. 

36.  The  Hydrostatic  Balance. — In   order  to  deter- 
mine the  specific  gravities  of  bodies  practically  and  with 
accuracy,  it  is  necessary  to   employ 
certain    instruments    for    weighing. 
These  are  the  Hi/drostaiic  Balance 
and  Hydrometers.* 

The  hydrostatic  balance  is  an  ordi- 
nary balance,  having  one  of  the  scale- 
pans  smaller  than  the  other,  and  at  a 
less  distance  from  the  beam ;  attached  fig.  26 

to  the  under  side  of  the  small  scale- 
pan  is  a  hook,  from  which  may  be  suspended  any  body  bv 
means  of  a  thin  platinum  wire,  horse-hair,  or  any  delicate 
thread.    The  body  whose  specific  gravity  is  to  be  found  is 

♦  Sometimes  called  Areometers. 


THE   COMMON  HYDROMETER.  77 

suspended  from  the  hook,  and  then  its  weight  is  determined. 
It  is  then  weighed  in  water,  and  thus  its  loss  of  weight  is 
ascertained,  wliich  is  the  weight  of  a  portion  of  water  equal 
in  volume  to  the  body. 

37.  The  Common  Hydrometer.— The  name  hydrom- 
eter is  given  to  a  class  of  instruments  used  for  determining 
the  specific  gravities  of  liquids  by  observing  either  the 
depths  to  which  they  sink  in  the  liquids  or  the  weights  re- 
quired to  make  them  sink  to  a  given  depth.  These 
instruments  depend  upon  the  principle  that  the 
weight  of  a  floating  body  is  equal  to  the  weight  of 
the  fluid  which  it  displaces. 

The  common  hydrometer  is  usually  made  of 
glass,  and  consists  of  a  straight  stem  ending  in  two 
hollow  spheres,  B  and  C,  the  lower  one  being 
loaded  so  as  to  keep  the  instrument  in  a  vertical 
position  Avhen  floating  in  the  liquid.     There  are  no 


weights  used  with  the  instrument;  but  the  stem       Fig. 27 
is  graduated,  so  as  to  enable  the  operator  to  ascer- 
tain the  specific  gravity  of  a  liquid  by  the  depth  to  which 
the  instrument  sinks  in  it. 

Let  Jc  =  the  area  of  a  section  of  the  stem,  r  :=  the  vol- 
ume, and  w  =  the  weight  of  the  hydrometer.  When  the 
hydrometer  floats  in  a  liquid  whose  specific  gravity  is  s,  let 
the  level  D  of  the  stem  be  in  the  surface  ;  and  when  it 
floats  in  a  liquid  whose  specific  gravity  is  s',  let  the  level  E 
be  in  the  surface.  Then  (Art.  34,  Eem.)  we  have  for  the 
weights  of  the  liquid  displaced  in  the  first  and  second  cases, 
respectively, 

w  =  s{v  —  ^'AD), 

w  =  s'  {v  —  k-AE}', 

but  the  weight  of  the  liquid  displaced  in  each  case  is  the 
same,  since  each  is  equal  to  the  weight  of  the  instrument. 


78  SIKES'S   HYDROMETER. 

i'  —  ^  —  ^'-AE 


(1) 


which  gives  the  ratio  of  the  specific  gravities  of  the  two 
liquids. 

Cor. — If  the  second  liquid  be  the  standard,  s  =  1,  and 
s,  the  specific  gravity  of  the  first  liquid,  is  given  in  (1). 

38.    Sikes's   Hydrometer.*— This  instrument  differs 
from  the  common  hydrometer  in  the  shape  of  the  stem, 
which  is  a  flat  bar  and  very  thin,  so  that  it  is 
exceedingly  sensitive.     It  is  generally  construct-  A ' 

ed  of  brass,  and  is  accompanied  by  a  series  of  E- 

small  weights  F,  which  can  be  slipped  over  the 
stem  above  C,  so  as  to  rest  on  C. 

The  weights  are  used  to  compensate  for  the 
great  sensitiveness  of  the  instrument,  which, 
without  the  weights,  would  render  it  applicable 
only  to  liquids  of  very  nearly  the  same  density. 

Let  ^-  =r  the  area  of  a  section  of  the  stem, 
V  =  the  volume,  and  iv  =  the  weight  of  the 
hydrometer.  When  the  instrument  floats  in  a  liquid  whose 
specific  gravity  is  s,  let  w'  =  the  weight  on  C  so  that  the 
level  D  of  the  stem  shall  be  in  the  surface  ;  and  when  it 
floats  in  a  liquid  whose  specific  gravity  is  s',  let  w"  =  the 
weight  on  C  so  that  the  level  E  shall  be  in  the  surface  ;  and 
let  v'  and  v"  be  the  volumes  of  id'  and  w".  Then  (Art.  34, 
Rem.)  we  have  for  the  weights  of  the  liquid  displaced  in  the 
first  and  second  cases,  respectively, 

w  +  w'  =  s  {v  +  v'  —  ^--AD), 

w  +  w"  =  s'  {v  +  v"  —  k-AE) ; 

(1) 

•  Besant's  Hydrostatics,  p.  127. 


w  +  tv' 

V  +  v" 

-  /t-AE 

to  -»-  to" 

V  -r  v' 

-  k-KY) 

mcBOLsoy's  htdkometer.  79 

Cou. — If  the  second  liquid  be  the  standard,  s  =  1,  and 
s,  the  specific  gravity  of  the  first  liquid,  is  given  in  (1). 

39.  Nicholson's  Hydrometer. — The  two  hydrometers 
just  described  are  used  for  obtaining  the  specific  gravities  of 
liquids.     Nicholson's  hydrometer  is  so  contrived  as 
to  determine  the  specific  gravity  of  solids  as  well        vA. 
as  liquids. 

It  consists  of  a  hollow  metallic  vessel  C,  gener- 
ally of  brass,  terminated  above  by  a  very  thin 
stem,  which  is  often  a  steel  wire,  bearing  a  small 
dish  A,  and  carrying  at  its  lower  end  a  heavy  cup 
D ;  on  the  stem  connecting  A  and  0,  a  well- 
defined  mark  B  is  made. 


^  C 


B 


(1)   Ih  determine  the  specific  gravity  of  a 
liquid. 

Let  to  be  the  weiglit  of  the  hydrometer,  w'  the  weight 
which  must  be  placed  in  the  dish  A,  in  order  to  sink  the 
stem  to  the  point  B  in  a  liquid  whose  specific  gravity  is  s, 
and  w"  the  weight  which  must  be  placed  in  the  dish  A,  to 
sink  the  stem  to  the  same  point  B  in  a  liquid  whose  specific 
gravity  is  s'.  Then  we  have  for  the  weights  of  the  liquid 
displaced  in  the  fii-st  and  second  cases,  respectively, 

IV  +  w'        and         tv  +  w"  ; 

and  since  the  volumes  displaced  are  the  same  in  both  cases, 
the  specific  gravities  are  as  the  weights  (Art.  29), 

s        w  +  w' 


W  +  20 


(1) 


Calling  the  second  liquid  the  standard,  s'  =  1,  and  (1) 
becomes 

lU  +  w    '  ^  ' 

which  is  the  specific  gravity  required. 


80  EXAMPLES. 

(2)  To  determine  the  specific  gravity  of  a  solid. 

Let  w  be  the  weight  which  must  be  placed  in  the  dish  A, 
to  sink  the  stem  to  the  point  B  in  a  liquid  whose  specific 
gravity  is  s. 

Put  the  solid  in  the  dish  A,  and  let  w'  be  the  weight 
which  must  be  added  to  the  solid  to  sink  the  stem  to  the 
point  B  in  the  same  liquid. 

Then  put  the  solid  in  the  lower  dish  D,  and  let  w"  be  the 
weight  required  in  the  upper  dish  A  to  siuk  the  stem  to  the 
point  B  in  the  same  liquid. 

Hence,  the  weight  of  the  solid  =  tv  —  ic',  and  its  weight 
in  the  liquid  =  tv  —  lu". 

Therefore  the  weight  lost,  which  is  tlie  weight  of  the 
liquid  displaced  by  the  solid  =  w"  —  w'.  Hence,  denoting 
by  S  the  specific  gravity  of  the  solid,  we  have 

8        w  —  w 


s        w    —  w 
If  the  liquid  is  the  standard,  s  =  1,  and  (3)  becomes 


(3) 


S  =  ^t^"-,,  (4) 

10     —  w  ' 


which  is  the  specific  gravity  required. 


KXAMPLES, 


1.  If  an  iceberg  whose  density  is  0.918  float  in  a  liquid 
whose  density  is  1.028,  what  is  the  ratio  of  the  part  sub- 
merged to  that  which  is  above  water  ?  Ans.  8.3  :  1. 

2.  How  much  of  its  weight  will  112  lbs.  of  iron  lose,  if 
immersed  in  water,  the  density  of  Jon  being  7.25  times 
that  of  water  ?  Ans.  15.448  lbs. 

3.  If  20  lbs.  of  cork  be  immersed  in  water,  with  what 
force  will  it  rise  towards  the  surface,  its  density  being  0.24 
times  that  of  water  ?  Atis.  63^  lbs. 


EXAMPLES.  81 

4.  If  a  piece  of  wood  whose  vertical  height  is  2  ft.  be 
placed  in  the  Dead  Sea,  how  many  inches  will  it  become 
submerged,  the  densities  of  the  wood  and  Dead  Sea  water 
being  .53  and  1.24  respectively  ?  A7is.  10.26  ins. 

5.  Find  the  depth  to  which  a  rectangular  block  will  sink 
in  water,  the  depth  of  the  block  being  a  feet,  and  the  weight 
of  each  cubic  foot  of  it  being  lo  lbs.  .         aw 

^''''  WE' 

6.  A  barge  of  a  rectangular  shape  is  Z  ft.  long,  b  ft.  broad, 
and  a  ft.  deep,  outside  measure.  The  thickness  of  the 
planking  is  e  ft.,  and  the  weight  of  a  cubic  foot  of  the  tim- 
ber is  w  lbs.  To  what  depth  will  the  barge  sink  when 
loaded  with  W  lbs.  ? 

w  [abl  —(a  —  e)  {h  —  2e)  {I  —  2e)]  +  W 


Ans. 


62.bbl 


7.  A  cylindrical  piece  of  wood,  weight  W,  floats  in  water 

with  its  axis  vertical  and  immersed  to  a  depth  h.     Find 

how  much  it  will  be  depressed  by  placing  a  weight  zo  on  the 

top  of  it.  i         w  , 

Ans.  -TT^h. 
W 

8.  An  isosceles  triangle  floats  in  water  with  its  base  hori- 
zontal. Find  the  position  of  equilibrium  when  the  base  is 
above  the  surface,  its  height  being  li  and  its  density  being 

■I  that  of  water.  ,        h    ,- 

'  Ans.  ^v6. 

o 

9.  A  rectangular  barge,  I  ft.  long,  b  ft.  broad,  and  a  ft. 
deep,  outside  measure,  sinks  to  \  its  whole  depth  when  un- 
loaded.    Required  its  weight  in  lbs.  Ans.  12.5abl. 

10.  If  a  rectangular  barge  sinks  to  ^  of  its  whole  depth 
when  unloaded,  and  to  f  of  its  whole  depth  when  loaded, 
find  the  load,  the  weight  of  the  barge  being  w.   Ans.  |w. 

11.  The  diameter  of  the  base  of  a  right  cone  is  2r,  its 
altitude  is  h,  and  its  density  is  f  that  of  water.     To  what 


82  EXAMPLES. 

depth  will  the  cone  sink  when  it  floats  with  its  vertex  down- 
wards ?  .  /*  8/TS 

A71S.  ;jvl8. 
o 

V2.  A  liemispherical  vessel,  whose  weight  is  10,  floats  upon 
a  fluid  with  ^  of  its  radius  below  the  surface.  What  weight 
must  be  put  into  the  vessel  so  that  it  may  float  with  f  of  its 
radius  below  the  surface ?  Ans.   |z/,'. 

13.  Let  the  pontoon  in  Ex.  4,  Art.  26,  be  a  cylinder, 
lengtii  I,  Avith  hemispherical  ends,  radius  r ;  to  find  the 
load  requisite  to  sink  the  pontoon  to  a  given  depth  u. 

Ans.  [Al  +  7ra2  (r  —  -^«)]  62.5, 
where  A  =  the  area  ADK  (Fig.  22). 

14.  Required  the  thickness  of  a  hollow  globe  of  copper 
whose  density  is  9  times  that  of  water,  so  that  it  may  just 
float  when  wholly  immersed  in  water,  r  being  the  exterior 
radius.  Ans.  r(l  — -fv^s). 

15.  A  cubical  box,  the  volume  of  which  is  one  cubic  foot, 
is  three-fourths  tilled  with  water,  and  a  leaden  ball,  the 
volume  of  which  is  72  cubic  inches,  is  lowered  into  the 
water  by  a  string.  It  is  required  to  find  the  increase  of 
pressure  (1)  on  the  base  and  (2)  on  a  side  of  the  box. 

Ans.  (1)  41|  oz. ;  (2)  32+  oz. 

16.  If  the  height  of  the  parallelopiped  in  Ex.  2,  Art.  28, 
is  0.9  of  the  breadth,  and  if  p  =  |,  find  the  inclination  d 
that  the  parallelopiped  may  float  in  indifferent  equilibrium. 

Ans.  e  =  33°  15'. 

17.  What  is  the  weight  of  a  cube  of  gold  whose  side  is 
3  ins.,  its  specific  gravity  being  19.35  ?     Ans.  18.896  lbs. 

18.  What  is  the  volume  of  a  piece  of  platinum  whose 
weight  is  10  lbs.,  its  specific  gravity  being  22.06? 

Ans.   12.533  en.  ins. 

19.  A  piece  of  lead,  whose  weight  is  511.65  grs.,  weighs 
in  water  466.57  grs.     Required  its  specific  gravity. 

Ans.   11.35. 


EXAMPLES.  83 

20.  A  sovereign,  whose  weight  is  123.02 'grs.,  weighs  in 
water  116.02  grs.     Required  its  specific  gravity. 

Ans.  17.574. 

21.  Find  the  specific  gravity  of  a  piece  of  wood,  whose 
weight  is  50  grs.  To  sink  it  we  attach  a  brass  weight 
wliich,  when  immersed  in  the  water,  weighs  87.22  gjs. ;  the 
weight  of  tlie  compound  body  when  immersed  is  42.88. 

Ans.  0.53. 

22.  A  piece  of  wood  weighs  4  lbs.  in  air  and  a  piece  of 
lead  weighs  4  lbs.  in  water ;  the  lead  and  wood  together 
weigh  3  lbs.  in  water.  Find  the  specific  gravity  of  the 
wood.  Ans.  0.8. 

23.  A  body  immersed  in  water  is  balanced  by  a  weight 
P,  to  which  it  is  attached  by  a  string  passing  over  a  fixed 
pulley  ;  when  half  immersed,  it  is  balanced  in  the  same  way 
by  a  Aveiglit  2F.     Find  the  specific  gravity  of  the  body. 

Ans.  f. 

24.  Find  the  weight  of  a  cubical  block  of  stone  whose 
side  is  4  ft.,  and  specific  gravity  1-^.  Ans.  80000  oz. 

25.  A  body  weighing  20  grs.  has  a  specific  gravity  of  2^. 
Required  its  weight  in  water.  Ans.   12  grs. 

26.  An  island  of  ice  I'ises  30  ft.  out  of  the  water,  and  its 
upper  surface  contains  |  of  an  acre.  Supposing  the  mass  to 
be  cylindrical,  required  (1)  its  weight,  and  (2)  depth  below 
the  water,  the  sj)ecific  gi-avity  of  sea-water  being  1.0263,  and 
that  of  ice  .92.  Ans.  (1)  242900  tons;  (2)  259.64  ft. 

27.  A  i)iece  of  wood  weighs  12  lbs.,  and  when  attached 
to  22  lbs.  of  lead  and  immersed  in  water,  tlie  whole  weighs 
8  lbs.  The  specific  gravity  of  lead  being  11,  required  that 
of  the  wood.  Ans.  \. 

'28.  A  solid  which  is  lighter  than  water  weighs  5  lbs., 
and  when  it  is  attached  to  a  piece  of  metal,  the  whole 
"weighs  7  lbs.  in  water.     The  weight  of  the  metal  in  water 


84  EXAMPLES. 

being  9  lbs.,  compare  the  specific  gravities  of  the  solid  and 
of  water.  Ans.  5  :  7. 

29.  A  piece  of  wood  which  weighs  57  lbs.  in  vacuo,  is  at- 
tached to  a  bar  of  silver  weighing  42  lbs.,  and  the  two 
together  weigh  38  lbs.  in  water.  Find  the  specific  gravity 
of  the  wood,  that  of  water  being  1,  and  that  of  silver  10.5. 

Ans.   1. 

30.  Equal  weiglits  of  two  fluids,  whose  specific  gravities 
are  s  and  2s,  are  mixed  togetiier,  and  one-third  of  the  whole 
volume  is  lost.  Find  the  specific  gravity  of  the  resulting 
fluid.  Ans.  2s. 

31.  Two  fluids  of  equal  volume,  and  of  specific  gravities 
s  and  2s,  lose  I  of  their  whole  volume  when  mixed  together. 
Find  the  specific  gravity  of  the  mixture.  Ans.  2s. 

32.  A  cylinder  floats  vertically  in  a  fluid  with  8  ft.  of  its 
length  above  the  fluid ;  find  the  whole  length  of  the  cylin- 
der, the  specific  gravity  of  the  fluid  being  three  times  that 
of  the  cvlinder.  Ans.  12  ft. 

33.  A  body  floats  intone  fluid  with  |  of  its  volume  im- 
mersed, and  in  another  with  |  immersed.  Compare  the 
specific  gravities  of  the  two  fluids.  Ans.  15  :  16. 

34.  A  block  of  wood,  the  volume  of  which  is  4  cubic  feet, 
floats  half  immersed  in  water.  Find  the  volume  of  a  piece 
of  metal,  the  specific  gravity  of  which  is  7  times  that  of  the 
wood,  which,  when  attached  to  the  lower  portion  of  the 
wood,  will  just  cause  it  to  sink.      Aiis.  f  of  a  cubic  foot. 

35.  A  cone,  whose  specific  gravity  is  |,  floats  on  the  water 
with  its  axis  vertical,  (1)  with  its  vertex  downwards  and  (2) 
with  its  vertex  upwards.  What  part  of  the  axis  is  immersed 
in  each  case?  Ans.  (1)  ^  ;  (2)  0.0436. 

36.  A  cone,  whose  specific  gravity  is  |,  floats  with  its 
axis  vertical.  Compare  the  portions  of  the  axis  immersed, 
(1)  when  the  vertex  is  u|)wards,  (2)  when  it  is  downwards. 

Ans.  'V^2-l:n., 


EXAMPLES.  85 

37.  A  block  of  ice,,  the  volume  of  which  is  a  cubic  yard, 
is  observed  to  float  with  -^^  of  its  volume  above  the  surface, 
and  a  small  piece  of  granite  is  seen  embedded  in  the  ice. 
Find  the  size  of  the  stone,  the  specific  gravities  of  ice  and 
granite  being  respectively  .918  and  2.65. 

Ans.  -g^  of  a  yard. 

38.  A  cylindrical  glass  cup  weighs  8  ozs.,  its  external 
radius  is  1^  ins.,  and  its  height  4^  ins.  If  it  be  allowed  to 
float  in  water  with  its  axis  vertical,  find  what  additional 
weight  must  be  placed  in  it,  in  order  that  it  may  sink. 

^^*-  (-^64"  -  8j  oz. 

39.  Find  the  position  of  equilibrium  of  a  cone,  floating 
with  its  axis  vertical  and  vertex  upwards,  in  a  fluid  of  which 
the  density  bears  to  the  density  of  the  cone  the  ratio  27  :  19. 

ns.  ^  of  the  axis  is  immersed. 

40.  The  whole  volume  of  a  hydrometer  is  5  cu.  ins.,  and 
its  stem  is  ^  of  an  inch  in  diameter;  the  hydrometer  floats 
in  a  liquid  A,  with  one  inch  of  the  stem  above  the  surface, 
and  in  a  liquid  B  with  two  ins,  above  the  surface.  Compare 
the  specific  gravities  of  A  and  B. 

Ans.  1280  —  TT  :  1280  —  27r. 

41.  What  volume  of  cork,  specific  gravity  .24,  must  be 
attached  to  6  lbs.  of  iron,  specific  gravity  7.6,  in  order  that 
the  whole  may  just  float  in  water  ? 

Ans.  x^^Vs"  *^f  ^  cubic  foot. 

42.  If  a  piece  of  metal  weigh  in  vacuum  200  grs.  more 
than  in  water,  and  160  grs.  more  than  in  spirit,  what  is  the 
specific  gravity  of  spirit  ?  Ans.  f. 

43.  A  piece  of  metal  whose  weight  in  water  is  15  ozs,,  is 
attached  to  a  piece  of  wood,  which  weighs  20  ozs,  in  vacuum, 
and  the  weight  of  the  two  in  water  is  10  ozs.  Find  the  spe- 
cific gravity  of  the  wood.  Ans.  -f. 


86  EXAMPLES. 

44.  A  crystal  of  salt  weighs  6.3  grs.  in  air;  when  covered 
with  wax,  the  specific  gravity  of  which  is  .96,  the  whole 
weighs  8.:^3  grs.  in  air  and  3.03  in  water.  Find  the  specific 
gravity  of  salt.  Ans.  1.9  nearly. 

45.  A  Nicholson's  Hydrometer  Aveighs  6  ozs.,  and  it  is 
requisite  to  place  weights  of  1  oz.  and  1^  ozs.  in  the  upper 
cup  to  sink  the  instrument  to  the  same  point  in  two  differ- 
ent liquids.     Compare  the  specific  gravities  of  the  liquids. 

Ans.  4  :  5. 

46.  A  diamond  ring  weighs  69|^  grs.,  and  64|^  grs.  in 
water.  The  specific  gravity  of  gold  being  16^,  and  that  of 
diamond  'd\,  what  is  the  weight  of  the  diamond  ? 

Ans.  3|^  grs. 

47.  A  body  A  weighs  10  grs.  in  water,  and  a  body  B 
weighs  14  grs.  in  air,  and  A  and  B  together  weigh  7  grs.  in 
water.  The  specific  gravity  of  air  being  .0013,  required  (1) 
the  specific  gravity  of  B,  and  (3)  the  number  of  grs.  of 
water  equal  to  it  in  volume. 

Ans.   (1)  .8337;  (3)  17.033  grs. 

48.  A  compound  of  gold  and  silver,  weighing  10  lbs.,  has 
a  specific  gravity  of  14,  that  of  gold  being  19.3,  and  that  of 
silver  being  10.5.  Eequired  the  Aveights  of  the  gold  and 
the  silver  in  the  compound. 

Ans.  Gold  =  5.483  lbs.;  silver  =  4.517  lbs. 

49.  A  diamond  ring  weighs  65  grs.  in  air  and  60  in  water. 
Find  the  weight  of  the  diamond,  if  the  specific  gravity  of 
gold  is  17.5,  and  that  of  the  diamond  3|^.    Ans.  6.875  grs. 

50.  The  crown  made  for  Hiero,  King  of  Syracuse  (Art. 
34,  note),  with  equal  weights  of  gold  and  silver,  were  all 
weighed  in  water  ;  the  crown  lost  ^  of  its  weight,  the  gold 
lost  -^  of  its  weight,  and  the  silver  lost  -^  of  its  Aveight. 
Prove  that  the  gold  and  silver  were  mixed  in  the  proportion 
of  11  :  9. 


EXAMPLES.  87 

51.  A  ring  consists  of  gold,  a  diamond,  and  two  equal 
rubies  ;  it  weighs  44J  grs.,  and  in  water  38|-  grs. ;  when  one 
ruby  is  taken  out,  it  weighs  2  grs.  less  in  water.  Find  the 
weight  of  the  diamond,  the  specific  gravity  of  gold  being 
lOJ,  of  diamond  3J,  of  ruby  3.  Ans.   b^  grs. 

52.  If  the  jsrice  of  pure  whiskey  be  $4  per  gallon,  and  its 
specific  gravity  be  .75,  what  should  be  the  price  of  a  mixture 
of  whiskey  and  water  which  on  gauging  is  found  to  be  of 
specific  gravity  . 8  ?  Ans.  $.3.20. 

53.  How  deep  will  a  paraboloid  sink  in  a  fluid  whose  spe- 
cific gravity  is  n  times  that  of  the  solid,  the  axis  being 
vertical  and  equal  to  a,  and  the  vertex  upwards  ? 

.            ^/n  —  a/w  —  1 
Ans.  a := • 

y/n 

54.  A  cubic  inch  of  metal,  whose  specific  gravity  is  m,  is 
formed  into  a  hollow  cone,  and  immersed  with  its  vertex 
downwards.  Determine  the  ratio  of  the  altitude  to  the 
exterior  radius  of  its  base,  when  the  surface  immersed  is  a 
minimum.  ,  j_ns.  V3. 


CHAPTER    III. 

EQUILIBRIUM    AND     PRESSURE     OF    GASES.  —  ELASTIC 

FLUIDS. 

40.  Elasticity  of  Gases.— The  pressure  of  an  elastic 
fluid  is  measured  exactly  in  the  same  way  as  the  pressure  of 
a  liquid  (Art.  6),  and  the  equality  of  pressure  in  every  direc- 
tion, and  of  transmission  of  pressure,  are  equally  true  of 
liquids  and  gases  (Arts.  7  and  8).  There  is,  however,  this 
difference  between  a  liquid  and  a  gas:  when  a  liquid  is  con- 
fined in  a  vessel,  no  pressure  is  exerted  against  the  sides 
except  that  which  is  due  to  the  weight  of  the  liquid  itself, 
or  that  which  is  transmitted  by  the  liquid  from  some  point 
on  the  surface  at  which  an  external  force  is  applied; 
whereas,  if  a  gas  be  contained  in  a  closed  vessel,  there  is, 
although  modified  by  the  action  of  gravity,  an  outward 
pressure  exerted  against  the  sides,  which  is  due  to  the  elas- 
ticity of  the  gas,  and  which  depends  upon  its  volume  and 
temperature.*  It  is  therefore  evident  that  generally  a  gas 
cannot  have  a  free  surface  like  a  liquid  (Art.  11),  for  such 
a  surface  implies  that  at  each  point  the  pressure  is  nothing, 
i.  e.,  if  it  be  covered  by  an  envelope  everywhere  in  close  con- 
tact with  it,  no  pressure  is  exerted  against  the  envelope.  It 
is  also  evident  that,  if  a  portion  of  the  gas  be  withdrawn 
from  the  vessel,  that  which  remains  will  not  fill  the  same 
part  of  the  vessel  that  it  occupied  before,  as  in  the  case  of  a 
liquid,  but  will  expand  so  as  to  fill  the  whole  vessel,  press- 
ing, but  with  diminished  force,  against  its  sides  at  every 
point  (Art.  2).  From  this  property  of  gases,  they  are  called 
elastic  fluids ;   the  outward  pressure  which   a  gas  exerts 

♦  If  the  gas  is  not  confined  within  a  limited  space,  the  effect  of  its  elasticity 
might  be  the  nnlimited  expansion  and  ultimate  dispersion  of  the  gas. 


PRESSURE   OF  THE  ATMOSPHERE.  89 

against  the  walls  of  the  vessel  enclosing  it  is  called  its  elas- 
tic force. 

The  action  of  a  common  syringe  will  serve  to  illustrate 
the  elasticity  of  atmospheric  air.  If  the  piston  be  drawn 
out,  and  the  open  end  of  the  syringe  then  closed,  a  consid- 
erable effort  will  be  required  to  force  in  the  piston  to  more 
than  a  small  part  of  the  length  of  its  range,  and  if  the 
syringe  be  air-tight  and  strong  enough,  it  will  require  the 
application  of  great  power  to  force  the  piston  down  through 
nearly  the  whole  of  its  range.  This  experiment  also  shows 
that  the  pressure  increases  with  the  compression,  the  air 
within  the  syringe  acting  as  an  elastic  cushion.  If  the 
piston  be  let  go,  after  being  forced  in,  it  will  be  driven 
back,  the  air  within  expanding  to  its  original  volume. 

An  inverted  glass  cylinder,  carefully  immersed  in  water, 
furnishes  another  simple  illustration  of  the  elasticity  of  air. 
Holding  the  cylinder  vertical,  it  may  be 
pressed  down  in  the  water  without  much 
loss  of  air,  and  it  will  be  seen  that  the  sur- 
face of  the  water  within  the  vessel  CD  is 
below  the  surface  of  the  water  outside  AB. 
It  is  evident  that  the  downward  pressure  of 
the  air  within  at  CD  is  equal  to  the  upward 
pressure  of  tiie  water  at  the   same  place,  F'g-so 

which  (Art.  11,  Cor.  2)  is  equal  to  the  pressure  on  the  np- 
j)er  surface  AB,  increased  by  the  pressure  due  to  the  depth 
of  the  surface  CD  below  the  upper  surface ;  hence  the  air 
within,  which  has  a  diminished  volame,  has  an  increased 
pressure. 

41.  Pressure  of  the  Atmosphere. — If  a  glass  tube* 
about  three  feet  in  length,  closed  at  one  end,  be  filled  with 
mercury,  and  then,  with  the  finger  pressed  to  the  open  end 


A 

C      D 

B 

g^— ~ 

sii^^ 

S 

^^^" 

^S 

*  This  experiment  was  first  made  by  Torricelli,  and  hence  is  called  TorricellVs 
Experiment,  and  the  vacant  space  above  the  mercury  in  the  tube  is  called  the  Torri- 
cellian Vacuum. 


90  WEIGHT  OP  THE  AIR. 

SO  as  to  close  it,  inverted  in  a  vessel  of  mercury  so  as  to  im- 
merse its  open  end,  it  will  be  found  on  removing  the  finger 
that  the  mercury  in  the  tube  will  descend  through  a  certain 
space,  leaving  a  vacuum  at  the  top  of  the  tube,  but  resting 
with  its  upper  surface  at  a  height  of  about  29  or  30  inches 
above  the  surface  of  the  mercury  in  the  vessel.  It  thus 
appears  that  the  atmospheric  pressure,  acting  on  the  sur- 
face of  the  mercury  in  the  vessel,  and  transmitted  (Art.  8), 
supports  the  column  of  mercury  in  the  tube,  and  hence  that 
the  weight  of  the  mercurial  column  is  exactly  equal  to  the 
weight  of  the  atmospheric  column  standing  on  an  area 
equal  to  that  of  the  internal  section  of  the  tube.  The 
Weight  of  this  column  of  mercury  then  is  an  exact  measure 
of  the  atmospheric  pressure,  or  of  the  elastic  force  of  the 
atmosphere  at  any  instant. 

42.  Weight  of  the  Air. — This  may  be  directly  proved 
by  weighing  a  flask  filled  with  air,  and  afterwards  weighing 
it  when  the  air  has  been  withdrawn  by  means  of  an  air- 
pump  ;  the  difference  of  the  weights  is  the  weight  of  the 
air  contained  by  the  flask. 

The  opinion  was  long  held  that  air  was  without  weight, 
or  rather,  it  never  occurred  to  any  of  the  philosophers  who 
preceded  Galileo  to  attribute  any  influence  in  natural  phe- 
nomena to  the  weight  of  the  air.  The  fact  that  air  has 
weight  escapes  common  observation  in  consequence  of  its 
extreme  levity  compared  with  solids  and  liquids,  and  espe- 
cially in  consequence  of  its  being  the  medium  by  which  we 
are  continually  surrounded.  The  experiment  of  weighing 
air  was  performed  successfully  for  the  first  time  in  1650,  by 
Otto  Guericke,  the  inventor  of  the  air-pump.* 

By  means  of  the  weight  of  air  we  may  account  for  the 
fact  of  atmospheric  pressure.  The  earth  is  surrounded  by 
a  quantity  of  air,  the  height  of  which  is  limited  (see  Art. 

*  Deschanel's  Natural  Philosophy,  Part  I.,  p.  141. 


THE  BAROMETER.  91 

72)  ;  and  if  we  suppose  a  cylindrical  column  extending 
above  any  horizontal  area  to  the  surface  of  the  atmosphere, 
the  weight  of  the  column  of  air  must  be  entirely  supported 
by  the  horizontal  area  upon  which  it  rests,  and  ths  pressure 
upon  the  area  is  therefore  equal  to  the  weight  of  the  column 
of  air.  The  pressure  of  the  air  must  then  diminish  as  the 
height  above  the  earth's  surface  increases  ;  and  from  exper- 
iments in  balloons  and  in  mountain  ascents,  this  is  found  to 
be  the  case. 

The  action  of  gravity  is  equivalent  to  the  effect  of  a  compression  of 
the  gas,  and  it  i8  thus  seen  that  the  pressure  of  a  gas  is  in  fact  caused 
by  its  weight,  as  in  the  case  of  a  liquid. 

Taking  tt  for  the  pressure  of  the  air  at  any  given  place 
(Art.  11,  Cor.  2),  and  assuming  that  the  density  of  the  air 
throughout  the  height  z  is  constant  and  equal  to  p,  the 
pressure  at  the  height  z  will  be 

TT  —  gpz.  (1) 

Cor. — It  may  be  shown,  in  the  same  manner  as  for  air, 
that  any  other  gas  has  weight,  and  that  the  intrinsic  weight 
is  in  general  different  for  different  gases.  Carbonic  acid 
gas,  for  instance,  is  heavier  than  air,  and  this  is  illustrated 
by  the  fact  that  it  can  be  poured,  as  if  it  were 
liquid,  from  one  jar  to  another. 


43.  The  Barometer.  —  This  instrument, 
which  is  employed  for  measuring  the  pressure 
of  the  atmosphere,  is,  in  its  simplest  form,  a 
straight  glass  tube  AB,  about  32  or  33  inches 
long,  containing  mercury,  and  having  its  lower 
end  immersed  in  a  small  cistern  of  mercury ;  the  ^.  ~-~ 
end  A  is  hermetically  sealed,  and  there  is  no  air 
in  the  branch  AB.  Since  the  pressure  of  a  fluid  at  rest  is 
the  same  at  all  points  of  the  same  horizontal  plane  (Art.  10), 
the  pressure  at  B,  in  the  interior  of  the  tube,  is  equal  to 


92  THE    WATER-BAROMETER. 

the  atmospheric  pressure  on  the  mercury  at  C,  which  is 
transmitted  from  the  surface  of  the  mercury  in  tiie  cistern 
to  the  interior  of  the  tube;  and  as  there  is  no  pressure  on 
the  surface  at  P,  it  is  clear  that  the  pressure  of  the  air  on  C 
is  the  force  which  sustains  the  column  of  mercury  PB. 

Let  a  be  the  density  of  mercury,  and  rr  the  atmospheric 
pressure  at  C ;  then  we  have 

TT  =  ^aPB,  (1) 

and,  since  g  and  a  are  constant,  the  height  PB  may  be  used 
as  a  measure  of  the  atmosplieric  pressure. 

44.  The  Mean  Barometric  Height.  —  The  mean 
height  of  the  barometric  column  at  the  level  of  the  sea  is 
found  to  vary  with  the  latitude,  but  it  is  generally  between 
29^  and  30  inches.  The  atmosphere  is  subject  to  continual 
changes,  some  irregular,  others  periodical.  If  the  density 
and  consequent  elastic  force  of  the  air  be  increased,  the  col- 
umn of  mercury  will  rise  till  it  reaches  a  corresponding 
increase  of  weight ;  if,  on  the  contrary,  the  density  of  the 
air  diminish,  the  column  will  fall  till  its  diminished  weight 
is  sufficient  to  restore  the  equilibrium.  The  barometric 
height  is  therefore  subject  to  continuous  variations;  during 
any  one  day  there  is  an  oscillation  in  the  column,  and  the 
mean  height  for  one  day  is  itself  subject  to  an  annual  oscil- 
lation, independently  of  irregular  and  rapid  oscillations  due 
to  high  winds  and  stormy  weather.  Usually  the  height  of 
the  column  is  a  maximum  about  9  A.  m.  ;  it  then  descends 
until  3  P.  M.,  and  again  attains  a  maximum  at  9  p.  m.* 

45.  The  Water-Barometer. — Mercury  possesses  two 
great  advantages  over  other  liquids,  which  has  led  to  its 
being  selected  above  all  others  for  use  in  barometric  instru- 
ments. The  first  advantage  of  mercury  is  that  it  does  not 
give  off  vapor  at  ordinary  temperatures.     If  it  did,  the  space 

*  Besant's  Hydrostatics,  p.  75. 


MANOMETERS.  93 

AP  above  the  mercury  would  be  filled  with  an  elastic  vapor, 
which  would  press  down  upon  the  column,  so  that  its  weight 
would  no  longer  be  a  measure  of  the  atmospheric  pressure, 
but  of  the  ditference  between  this  pressure  and  the  elastic 
force  of  the  vapor  given  off.  The  second  advantage  is  that, 
on  account  of  the  great  density  of  mercury,  the  height  of 
the  column  which  measures  the  atmospheric  pressure  is  so 
small  that  barometers  constructed  with  it  are  of  a  very  con- 
venient size.  The  pressure  of  the  air  may  be  measured  by 
using  any  kind  of  liquid.  The  density  of  mercury  is  about 
13.595  times  that  of  Avater,*  and  therefore,  if  water  were 
used,  it  would  be  necessary  to  have  a  tube  of  great  length, 
since  the  column  of  water  in  the  water-barometer  would  be 
about  33f  feet. 

In  order  to  measure  easily  and  correctly  the  barometric 
height,  an  accurately  graduated  scale  is  added,  which  can 
be  moved  along  the  tube. 

Rem. — The  instrument  above  described  involves  the  essen- 
tial parts  of  a  barometer ;  it  is  the  province  of  Physics  to 
give  a  full  description  of  different  kinds  of  barometers, 
to  explain  their  use,  etc. 

46.  Manometers. — Barometers  are  used  not  only  to 
measure  the  pressure  of  the  external  air,  but  also  to  deter- 
mine the  elastic  force  of  gases  or  vapors  which  are  enclosed 
in  vessels.  When  thus  used,  they  are  called  manometers. 
These  instruments  are  filled  with  mercury,  and  are  either 
open  or  closed  ;  in  the  latter  case,  there  may  be  air  above  the 
column  of  mercury  or  there  may  be  a  vacuum.  The  manom- 
eter witli  a  vacuum  above  the  column  of  mercury  is  like  the 
common  barometer.  In  order  to  measure  with  it  the  elastic 
force  of  the  gas  or  vapor,  it  will  be  necessary  to  establish  a 
free  connection  between  the  cistern  of  the  barometer  and 
the  vessel  containing  the  fluid.     This  is  done  by  means  of  a 

*  Enc.  Brit.,  Vol.  XVI.,  p.  33, 


We 

i   Fio 


32 


9i  TEE  ATMOSPHERIC  PRESSURE. 

tube  DE,  one  end  of  which  E  opens  into  the  vessel  contain- 
ing the  fluid,  and  the  otlier  end  D  enters  above  the  level  of 
the  mercury  B  in  the- cistern.  By  this  means 
the  gas  from  the  vessel  flows  through  the  tube 
ED  into  the  cistern,  and  presses  a  column  of 
mercury  into  the  tube  AB,  the  height  of  which 
measures  the  elastic  force  of  the  gas  or  vapor  in 
the  vessel. 

When  the  elastic  force  of  the  fluid  is  consid- 
erable, it  is  usual  to  estimate  it  as  so  many 
atmospheres  :  for  instance,  steam,  in  the  boiler 
of  an  engine,  having  a  pressure  of  two  atmos- 
pheres, signifies  that  its  elastic  force  would  sus- 
tain a  column  of  about  60  inches  of  mercury. 
If  it  is  said  to  have  a  pressure  of  6  atmospheres,  it  means 
that  its  elastic  force  would  sustain  a  column  of  about  180 
inches  of  mercury  ;  and  so  on. 

47.  The  Atmospheric  Pressure  on  a  Square  Inch. 

— This  may  be  found  at  once  by  observing  that  it  is  the 
weight  of  a  cyUndrical  column  of  mercury  whose  base  is  a 
square  inch,  and  whose  height  is  equal  to  that  of  the 
barometric  column. 

Since  the  specific  gravity  of  mercury  is  13.595,  that  of 
water  being  1,  it  follows  that  the  pressure  of  the  air  on  a 
square  inch,  taking  30  inches  as  the  height  of  the  barometer 
at  the  sea  level, 

=  (30  X  13.595  X  62.5 -f- 1728)  lbs. 
=  14.7  lbs., 

and  this  is  called  the  pressure  of  one  atmosphere. 

ScH. — This  pressure  varies  from  time  to  time,  but  is  gen- 
erally between  14|^  and  15  lbs.  The  standard  usually 
adopted  where  the  English  system  of  measure  is  used  is 
14.7  lbs.  upon  the  square  inch,  which  corresponds  to  a  col- 


BOTLE  AND  MARIOTTE'S  LAW.  95 

amn  of  mercury  about  30  (exactly  29.922)  inches,  and  to  a 
column  of  water  about  34  (exactly  33.9)  feet  high.  A  }3ress- 
ure  of  two  atmospheres,  therefore,  would  mean  a  pressure 
of  29.4  lbs.  on  each  square  inch,  and  a  pressure  of  six  atmos- 
pheres would  mean  a  pressure  of  88.2  lbs.  on  each  square 
inch.     (See  Weisbach's  Mechs.,  p.  777.) 

EXAMPLES. 

1.  If  the  elastic  force  of  a  gas  is  2|^  atmospheres,  find  its 
pressure  in  lbs.  on  each  square  inch.         •  Ans.  36.75  lbs. 

2.  If  the  elastic  force  of  steam  in  a  boiler  be  b\  atmos- 
pheres, find  the  pressure  on  a  safety-valve  whose  area  is 
5.4  sq.  ins.  Ans.  436.59  lbs. 

48.  Boyle  and  Mariotte's  Law.*— Gases  readily  con- 
tract into  smaller  volumes  when  compressed.  "When  a  gas 
is  compressed,  its  elastic  force  is  increased  ;  and  when  it  is 
allowed  to  expand,  its  elastic  force  is  diminished.  The 
statement  of  the  law  which  expresses  the  relation  between 
the  pressure  and  the  volume,  or  the  pressure  and  the  density, 
of  gases  is  the  following  : 

The  pressure  of  a  given  quantity  of  air, 
at  a  given  temperature,  varies  inversely 
as  its  volume,  and  directly  as  its  density. 

Let  ABCD  be  a  bent  glass  tube,  the  shorter 
branch  of  which  can  have  its  end  D  closed, 
and  both  branches  being  vertical.  Let  a  little 
mercury  be  poured  in  at  A,  and  let  it  stand  at 
the  same  level  EF  in  both  branches.  Now 
close  the  end  D;  a  definite  volume  of  air  is  Fig.  33 

thus  enclosed  in  DE  under  a  pressure  equal  to 
that  of  the  external  air,  i.  e.,  the  elastic  force  of  the  enclosed 
air  DE  is  equal  to  the  atmospheric  pressure  exerted  on  F  in 


A 

L 

[ 

M 
H-" 

) 

N 

E 

F 

a 

B 

♦  The  experimental  proof  of  this  law  was  discovered  about  the  same  time  in 
England  by  the  Hon.  Robert  Boyle,  and  in  France  by  Mariotte. 


96  BOYLE  AND  MARIOTTE'S  LAW. 

the  open  branch,  and  is  therefore  equal  to  one  atmosphere 
(Art.  47). 

Take  DH  =  |DE,  and  pour  mercury  slowly  into  tiie 
tube  AB  till  it  stands  at  H  in  the  shorter  branch  ;  in  the 
longer  branch  it  will  be  found  to  stand  at  the  height  LK  = 
30  inches  above  IIK,  i.e.,  the  me/cury,  rising  in  the  shorter 
branch,  compresses  the  air  which  it  drives  before  it,  and 
when  the  air  in  the  shorter  branch  is  reduced  to  half  its 
volume,  its  elastic  force  or  pressure  is  two  atmospheres, 
since  it  now  sustains  not  only  the  atmospheric  pressure 
wiiich  is  exerted  on  the  surface  of  the  mercury  in  the  open 
branch,  but  also  the  weight  of  a  column  of  mercury  30 
inches  high.  When  mercury  is  poured  into  the  tube  till  it 
rises  in  the  shorter  branch  to  M,  where  DM  =:  -^DE,  it  w-ill 
be  found  to  stand  in  the  longer  branch  at  the  height  AN  = 
60  inches  above  MN,  i.  e.,  when  the  air  in  the  shorter 
branch  is  reduced  to  one-third  of  its  volume,  its  elastic 
force  or  pressure  is  three  atmospheres,  since  it  now  sustains 
the  atmospheric  pressure  and  the  weight  of  a  column  of 
mercury  60  inches  in  height.  In  the  same  way,  it  may  be 
shown  that  if  the  air  occupy  one-fourth  of  its  original  vol- 
ume DE,  it  will  sustain  a  pressure  of  four  atmospheres,  and 
so  on  for  any  number.  Hence,  generally,  the  2)resswe  of  a 
quantity  of  air  varies  inversely  as  its  volume. 

When  the  volume  is  reduced  to  one-half,  the  density  is 
doubled  ;  when  reduced  to  one-third,  the  density  is  trebled, 
and  so  on ;  that  is,  the  volume  varies  im^ersely  as  the  density. 
Hence,  the  pressure  varies  directly  as  the  density. 

Let  V  and  v'  be  the  volumes  of  a  given  mass  of  air,  p  and 
p'  the  corresponding  pressures,  and  p  and  p'  the  correspond- 
ing densities.     Then  we  have 

p'        V   ~  p"  ^  ^ 

p  =  hp,  (2) 

where  h  is  a  constant  to  be  determined  by  experiment. 


EXAMPLES.  97 

Rem.  1. — It  has  been  shown  by  a  series  of  experiments 
that  this  law  connecting  the  elastic  force  and  volume  of  a 
gas  under  a  constant  temperature  is  sensibly  true  for  air  and 
most  gases  as  far  as  a  pressure  of  100  atmospheres.*  It  is 
only  when  the  pressures  are  very  great  that  variations  from 
the  law  are  observed,  and  even  then  the  departure  from  the 
law  is  but  small,  especially  with  those  gases  which  we  are 
not  able  to  condense  into  liquids.  With  gases  which  undergo 
liquefaction  at  moderate  pressures,  the  departure  from  the 
law  is  greater,  and  increases  as  the  state  of  liquefaction  is 
approached,  f 

Rem.  3. — In  conducting  this  experiment,  care  must  be 
taken  to  have  the  temperatures  the  same  at  the  beginning 
and  at  the  conclusion,  as  the  elastic  force  of  a  gas  under  a 
given  volume  is  influenced  by  changes  of  temperature.  For 
this  reason,  it  is  necessary  to  pour  in  the  mercury  gradually, 
and  to  allow  some  time  to  elapse  before  the  difference  of 
levels  is  observed,  si"  ''3,  whenever  a  gas  is  compressed,  an 
elevation  of  temperature  is  produced.  Therefore,  whatever 
heat  is  developed  by  increase  of  pressure  must  be  allowed  to 
pass  off  before  the  volume  of  gas  is  observed. 

EXAMPLES. 

1.  Let  DE  (Fig.  33),  be  10  inches  ;  if  mercury  be  poured 
in  until  the  level  in  the  closed  branch  stands  3  inches  above 
EF,  and  in  the  open  branch  15.G4  inches,  find  the  elastic 
force  of  the  air  in  the  closed  branch,  the  barometer  standing 
at  29.5  inches. 

Since  the  levels  of  the  mercury  in  the  two  branches  stand 
at  15.64  and  3  inches,  the  level  in  the  longer  branch  is 
12.64  inches  above  that  in  tiie  closed  branch;  the  elastic 
force  of  the  compressed  air,  therefore,  sustains  a  column  of 


*  Galbraith's  Hydrostatics,  p.  .35. 

t  Weisbach's  Mechs.,  p.  782 ;  also  Twisden's  Mechs.,  p.  229. 


98  EFFECT  OF  HEAT  ON  GASES. 

mercury  12.64  inches  high,  together  with  the  atmospheric 
pressure,  which  by  the  barometer  is  shown  to  be  equal  to  a 
column  of  29.5  inches;  hence  the  elastic  force 

=  (12.64  +  29.5)  inches  =  42.14  inches. 

2.  If  the  level  in  the  closed  branch  rise  6.4  inches,  find 
the  height  to  which  the  level  in  the  open  branch  should 
rise,  the  barometer  standing  at  30.42  inches,  and  DE  being 
10  inches.  Ans.  60.48  inches. 

49.  Effect  of  Heat  on  Gases. — When  a  given  quan- 
tity of  air  or  gas  is  increased  in  temperature,  it  is  found 
that,  if  the  air  or  gas  cannot  change  its  volume,  its  elastic 
force  is  increased ;  but  if  the  air  can  expand  freely,  while 
its  elastic  force  remains  the  same,  its  volume  will  be  in- 
creased. 

To  illustrate  this,  take  an  air-tight  piston  in  a  vertical 
cylinder  containing  air,  and  let  it  be  in  equilibrium,  the 
weight  of  the  piston  being  supported  by  the  cushion  of  air 
beneath  it.  Eaise  the  temperature  of  the  air  in  the  cylinder 
by  immersing  it  in  hot  water;  (1)  the  piston  will  rise  in  the 
cylinder  as  the  volume  of  the  heated  air  expands  ;  and  when 
the  air  has  reached  the  temperature  of  the  surrounding 
water,  the  piston  will  cease  to  ascend,  and  will  remain  sta- 
tionary. But  (2)  if  we  suppose  that  when  the  heat  is 
applied,  the  piston  is  held  down  so  as  to  keep  the  air  under 
a  constant  Yolume,  an  effort  will  be  required  to  prevent  the 
piston  from  ascending  in  the  tube,  which  becomes  greater 
in  proportion  as  the  air  is  heated.     Hence 

(1)  The  effect  of  heat  on  a  given  quantity  of  air,  the 
elastic  force  remaining  constant,  is  to  expand  its 
volume. 

(2)  TJie  effect  of  heat  on  a  given  quantity  of  air,  the 
volume  remaining  constant,  is  to  increase  its  elastic 
force. 


THERMOMETERS.  99 

50.  Thermometers. — As  a  general  rule,  bodies  expand 
under  the  action  of  heat,  and  contract  under  the  action  of 
cold,  and  the  only  method  of  measuring  temperatures  is  by 
observing  the  extent  of  the  expansion  or  contraction  of 
some  known  substance.  Anybody  which  indicates  changes 
of  temperature  may  be  called  a  thermometer. 

As  the  expansions  of  different  substances  are  not  exactly 
proportional  to  one  another,  it  is  necessary  to  select  some 
one  substance  or  combination  of  substances  to  furnish  a 
standard,  and  the  standard  usually  adopted  for  all  ordinary 
temperatures  is  the  apparent  expansion  of  mercury  in  a 
graduated  glass  vessel ;  for  very  high  temperatures,  a  metal 
of  some  kind  is  the  more  useful,  and  for  very  low  tempera- 
tures, at  which  mercury  freezes,  alcohol  must  be  employed. 

TJte  memirial  thermometer  is  formed  of  a  thin  glass  tube 
of  uniform  bore,  terminating  in  a  bulb,  and  having  its  upper 
end  hermetically  sealed.  The  bulb  contains  mercury,  which 
also  extends  partly  up  the  tube,  and  the  space  between  the 
mercury  and  the  top  of  the  tube  is  a  vacuum.  Since  the 
glass,  as  well  as  the  mercury,  expands  with  an  increase  of 
temperature,  the  apparent  expansion  is  the  difference  be- 
tween the  actual  expansion  and  the  expansion  of  the  glass. 
The  construction  of  an  accurate  mercurial  thermometer  is 
an  operation  of  great  delicacy. 

In  Fahrenheit's  Tliermometer,  which  is  chiefly  used  in 
J^ngland  and  in  this  country,  the  freezing  point  is  marked 
32°,  and  the  boiling  point  212°.  The  space,  therefore,  be- 
tween these  two  points  is  180°. 

In  the  Centigrade  Thermometer  the  freezing  point  is 
marked  0°,  and  the  boiling  point  100°,  the  space  between, 
being  divided  into  100°. 

In  Reaumur's  Thermometer  the  freezing  point  is  also 
marked  0°,  but  the  boiling  point  is  marked  80°*. 

•  The  tetnperatnre  Indicated  by  the  boiling  point  is  the  same  in  all. 


100  EXAMPLES. 

Rej'. — Mercury  freezes  at  a  temperature  of  —40°  C.  or  F.,  and  boils 
at  a  temperature  of  about  350"  C.  or  663"  F.  ;  it  is  therefore  necessary, 
for  very  high  or  very  low  temperatures,  to  employ  other  substances. 

For  very  low  temperatures,  spirit  of  wine  is  used ;  this  liquid  has 
never  congealed,  although  a  temj>erature  of  —140'  C.  has  been  ob- 
served, which  is  the  lowest  temperature  yet  attained.* 

High  temperatures  are  compared  by  observing  the  expansion  of  bars 
of  metal  or  other  solid  substances,  and  instruments  called  pyrometers 
have  been  constructed  for  this  purpose. 

51.  Comparison  of  the  Scales  of  these  Thermom- 
eters.— Any  degrees  of  temperature  by  either  thermometer 
may  be  converted  into  the  corresponding  degrees  of  the 
other  thermometers  ;  for  the  space  between  the  fixed  points 
in  Fahrenheit's  being  180°,  in  the  Centigrade  100°,  and  in 
Reaumur's  80°,  we  have  180°  Fahrenheit  =  100°  Centi- 
grade =  80°  Reaumur  ;  and  therefore  each  of  Fahrenheit's 
degrees  =  -f  of  one  of  Centigrade  =  ^  of  one  of  Reaumur. 

Let  F,  C,  and  R  be  the  numbers  of  degrees  marking  the 
same  temperature  on  the  respective  thermometers ;  then 
since  the  space  between  the  boihng  and  freezing  points 
must  in  each  case  be  divided  in  the  same  proportion  by  the 
mark  of  any  given  temperature,  we  must  have 

F--ji2  _  _C   _  B^, 
180      ~"  100  ""  80 ' 
F—32        C        R 
—9-  =  5  =  T-  (^) 

Rem. — The  various  scales  were  fonued  in  the  early  part  of  the  18th 
century — Fahrenlieit's  in  1714,  at  Dantzic ;  Reaumur's  in  1731 ;  and 
the  Centigrade  somewhat  later.f 

EXAMPLES. 

1.  What  temperatures  on  the  other  two  scales  are  equiva- 
lent to  the  temperature  50°  F.  ?  |      Ans.  10°  C,  or  8°  R. 

*  Maxwell  on  Heat.  +  Besani's  Hydrostatics,  p.  88. 

X  It  is  usual,  in  stating  temperatures,  to  indicate  the  scale  referred  to  by  the  in> 
tials  F.,  C,  K. 


D ALTON'S  AND    GAY-LUSSAC'S  LAW.  101 

2.  Find  (1)  what  temperature  C.  is  the  same  as  60"  R., 
and  (2)  what  temperature  R.  is  the  same  as  45°  C. 

Ans.  (1)  75°  C;  (2)36°  R. 

52.  Expansion  of  Mercury.— The  expansion  of  mer- 
cury is  very  nearly  uniform  between  0°  and  300°.  Experi- 
ments show  that,  for  an  increase  of  1°  Centigrade,  the 
expansion  of  mercury  is  -^j-^,  or  .0001815  of  its  volume  ;* 
hence,  if  Of  be  the  density  at  a  temperature  t,  and  o^  the 
density  at  a  temperature  0°,  we  have 

<T„  =  <T^(1  +  .0001815/); 

or,  if  we  put  .0001815^  6,  we  have 

a,  =  a,  (1  -f  dt),  (1) 

which,  in  (1)  of  Art.  43,  gives 

Tr  =  </a,PB  =  (/a„(l_0/)PB,  (2) 

by  means  of  which  the  atmospheric  pressure  at  any  place 
can  be  calculated. 

53.  Dalton's  and  Gay-Lussac's  Law  of  the  Ex- 
pansion of  Gases  by  Heat. — The  following  experimental 
law  was  discovered  by  Gay-Lussac  f  and  Dalton,  and  more 
recently  corrected  by  Regnault. 

//  tlie  pressure  remains  constant,  an  increase  of 
temperature  of  1°  C.  produces  in  a  given  mass  of  air 
an  expansion  of  .003665  of  its  volume. 

By  means  of  this  experimental  law,  combined  w4th  Boyle's 
(Art.  48),  the  relation  between  the  pressure,  density,  and 
temperature  of  a  given  mass  of  air  or  gas  mav  be  expressed. 

Conceive  that  a  mass  of  air  at  the  temperature  of  0°  C.  is 
inclosed  in  a  cylinder  by  a  piston  to  which  a  given  force  is 

•  Enc.  Brit.,  Vol.  XVI.,  p.  33.  t  See  Deschanel's  Nat.  Phil.,  p.  807. 


102  V ALTON'S  AND    GAF-LUSSAC'S  LAW. 

applied;  let  the  temperature  be  iiicrearicd  to/;  the  piston 
will  then  be  forced  out  until  the  original  volume  t\  is  in- 
creased by  .003665/'?'o,  where  v^  is  the  volume  of  air  at  0°. 

Let  V  be  the  volume  of  the  same  mass  of  air  at  the  tem- 
perature / ;  then  we  have 

V  =  v^{l  +  .0036650  ; 

or,  denoting  .003665  by  «,  we  have 

V  =  V,  (1   +  at).  (1) 

Cor.  1. — If  Fahrenheit's  scale  is  used,  the  number  of  de- 
grees above  the  freezing  point  is  t  —  32  ;  and,  since  180°  F. 

correspond  to  100°  C,  the  expansion  for  1°  F.  is  '-^^^r  = 

loO 

;f^2^of  the  volume  at  32°  F.     The  more  accurate  value  of 

the  denominator  is  491.13. 

Hence,    the  increase  of  volume  =    **    ,._ —  ; 

492 

and,  for  the  whole  volume,  we  have 


V  =  v„  + 


492 


460  +  /  .... 

or,  V  =  v^  — 492~  '  <^) 

where  t  is  the  temperature  on  Fahrenheit's  scale,  and  z'^  is 
the  volume  at  32°  F. 

Cor.  2. — If  v'  be  the  volume  which  the  same  mass  of  air 
assumes  at  the  temperature  t',  we  have 

460  +  t' 


Dividing  (3)  by  (2),  we  have 


(3) 


460  +  f  ,,. 

^=^460TT-  ^'^ 


PRESSURE,    TEMPERATURE,   AND   DENSITY.  103 

By  means  of  (4)  we  may  determine  the  volume  which  a 
gas  will  assume  at  a  given  temperature;  or,  conversely,  the 
temperature  it  will  have  under  a  given  volume,  if  the  volume 
it  has  at  any  given  temperature  is  known,  the  pressure  re- 
maining constant. 

EXAMPLES. 

1.  If  100  cubic  inches  of  gas  at  68°  F.  be  heated  to 
130°  F.,  find  the  volume,  the  pressure  being  constant. 

Ans.  109.85  cu.  ins. 

2.  A  mass  of  air  at  50°  F.  is  raised  to  51°  F.  What  is 
the  increase  of  its  volume  under  a  constant  pressure  ? 

Ans.  1^0^  its  volume. 

54.  Law  of  the  Pressure,  Temperature,  and 
Density  of  a  Mass  of  Gas. — Let  p,  p,  and  v  be  the 

pressure,  density,  and  volume  of  a  mass  of  gas  at  the  tem- 
perature /,  vo  and  po  the  volume  and  density  at  0°. 

Then,  when  p  remains  constant,  we  have,  from  (1)  of 
Art.  53, 

V  =  r,  (1  +  a/).  (1) 

Now,  if  t  remains  constant  while  the  gas  is  compressed 
from  V  to  v^,  the  volume  varies  inversely  as  the  density 
(Boyle's  Law)  ;  that  is, 

V  :  v^  ::  p^  :  p, 

which  in  (1)  gives, 

p^  =p(l  +  at).  (2) 

Substituting  in  (2)  of  Art.  48,  we  have 

p  =  kpf,  =  kp  (1  +  at).  (3) 

Cor.  1. — If  //,  p'  be  the  pressure  and  density  of  the  same 
gas  at  a  temperature  t',  we  have 

p'  =  kp'{l  +  at')',     ' 


104  PRESSURE,    TEMPERATURE,    AND  DENSITT. 

p    _p   1  +  ctt 


(4) 


Cor.  2. — If  the  volume,  aud  therefore  the  density,  re- 
mains constant,  while  the  temperature  rises,  the  pressure 
will  also  rise. 

Let  jB„  be  the  pressure  when  /  =  0,  v  and  p  remaining 
constant.     Then  (3)  becomes, 

Po  =  ^P'  (5) 

Substituting  in  (3),  Ave  have 

p=p^{l  +  at),  (6) 

where  p  and  jo^  are  the  pressui-es  at  the  temperatures  t  and 
0,  the  volume  being  constant. 
Let  t  =  1,  then  (6)  becomes 

p  —j)o  =  p^fi  —  .003665^0  (^rt-  53) ; 

that  is,  if  the  volmne  of  a  mass  of  gas  remains  con- 
stant, an  increase  of  temperature  of  1°  C.  produces  an 
increase  of  pressure  equal  to  .003665  of  its  original 
pressure. 

Cor.  3.— If  Fahrenheit's  scale  is  used,  (3),  (4),  and  (6) 
become  respectively 

p__P_  460  +  ^ 

y-p'460-4-r'  ^' 

460  +  t  ... 

Cor.  4. — If  jo'  be  the  pressure  of  the  same  gas  at  a  temper- 
ature /',  the  volume  remaining  constant,  we  have,  from  (9), 

,  _       460  +  t' 
P   -P^      493     ' 


ABSOLUTE   TEMPERATURE.  105 

...   Z  =  l^+A  no) 

p'        460  +  t"  ^^"^ 

in  which  p  and  p  are  the  pressures  corresponding  to  the 
temperatures  t  and  /'  of  a  given  mass  of  gas,  the  vohime 
being  constant. 

Cor.  5. — Since  the  volume  of  a  given  mass  of  air  varies 
inversely  as  its  density,  we  have,  from  (4)  and  (8), 

,'  =  ,\±^JL,  (11) 

1  +  at    p  ^     ' 

460  +  t'p 

where  v'  and  v  denote  the  volumes  of  a  given  mass  of  air  at 
the  temperatures  t'  and  t. 

EXAMPLES. 

1.  If  the  pressure  of  a  given  mass  of  gas  be  29.25  inches, 
at  the  temperature  56°  F.,  what  will  it  become  if  heated  to 
300°  F.,  the  volume  being  constant?   Ans.  43.081  inches. 

2.  If  200  cubic  inches  of  gas  at  60°  F.,  under  a  pressure 
of  30  inches  of  mercury,  be  raised  in  temperature  to  280°  F., 
while  the  pressure  is  reduced  to  20  inches,  find  the  volume. 

Ans.  426.9  cubic  inches. 

55.  Absolute  Temperature. — If  we  can  imagine  the 
temperature  of  a  gas  lowered  until  its  pressure  vanishes, 
without  any  change  of  volume,  we  arrive  at  what  is  called 
the  absolute  zero  of  temperature,  and  absolute  temperature 
is  measured  from  this  point.*         " 

Let  t^  represent  this  temperature  on  the  Centigrade 
scale;  then  (3)  of  Art.  54  becomes 

*  Besant^s  Hydromechanics,  p.  113. 


106  PRESSURE   OF  A   MIXTURE   OF  GASES. 

0  =  kp  (1  +  at,),  (1) 

or,  t.  =  --  =  -  273°. 

In  Fahrenheit's  scale,  the   reading  for  absolute  zero  is 
—  459°. 
Combining  (1)  of  this  Art.  with  (3)  of  Art.  54,  we  have 

p  =  kpa  [t  —  t,) 

=  Tcpa  {t  +  273)  =  hpaT,  (2) 

where  T  \q  the  absolute  temperature. 

If  V  and  p  be  the  volume  and  density  of  a  mass  of  gas,  pv 

is  constant,  and  therefore,  from  (2),  ^  is  constant ;  from 

which  it  appears  that  the  product  of  the  pressure  and 
volume  of  a  given  mass  of  gas  is  proportional  to  the 
absolute  temperature. 

ScH. — If  the  difference  of  temperature  between  tlie  freez- 
ing and  boiling  points  be  divided  into  a  hundred  degrees, 
as  in  the  Centigrade  thermometer,  the  freezing  point  will 
then  be  273°  and  the  boiling  point  373°  absolute  tempera- 
ture, and  the  zero  of  the  scale  will  be  that  temperature  at 
which  the  pressure  vanishes.  Denoting  the  absolute  tem- 
perature by  T,  and  the  ordinary  Centigrade  temperature  by 
t,  we  have 

T  =  273°  -I-  t.  (3) 

56.  The  Pressure  of  a  Mixture  of  Gases If  two 

liquids,  which  do  not  act  chemically  on  each  other,  are 
mixed  together  in  a  vessel  which  remains  at  rest,  they  will 
gradually  separate,  and  finally  attain  equilibrium  with  the 
lighter  liquid  above  the  heavier.  But  if  two  gases  are 
placed  in  communication  with  each  other,  even  if  the 
heavier  be  below  the  lighter,  they  will  rapidly  intermingle 


MIXTURE    OF  GASES.  107 

until  the  proportion  of  the  two  gases  is  the  same  throughout, 
and  the  greater  the  difference  of  density  the  more  , rapidly 
will  the  mixture  take  place. 

Take  two  different  gases,  of  the  same  temperature  and 
pressure,  contained  in  separate  vessels ;  let  a  communica- 
tion be  established  between  the  vessels,  and  it  will  be  found 
that,  unless  a  chemical  action  take  place,  tlie  two  gases  will 
permeate  each  other  till  they  are  completely  mixed,  and 
that,  when  equilibrium  is  attained,  the  pressure  of  the  mix- 
ture will  be  the  same  as  before,  provided  the  temperature  is 
the  same.  Hence,  from  this  experimental  fact,  the  follow- 
ing proposition  can  be  deduced. 

57.  Mixture  of  Equal  Volumes  of  Oases  having 
Unequal  Pressures. — //  tivo  gases  having  the  same 
temperature  he  mixed  together  in  a  vessel  of  volume  v, 
and  if  the  pressures  of  the  gases  luhen  respectively  con- 
tained in  V,  at  the  same  temperature,  he  p  and  p' ,  the 
pressure  of  the  mixture  will  he  p  +p'. 

Suppose  the  gases  are  separate.  Take  the  gas  whose 
pressure  is  jo,  and  change  its  volume  until  its  pressure  is  p', 
its  temperature  remaining  the  same.     Its  volume  will  then 

be,  by  Mariotte's  law  (Art.  48),  —7  v.  ^ 

Now  let  the  two  gases  be  mixed  without  change  of  vol- 
ume, so  that  the  volume  of  the  mixture  is 

.    P  P  +P' 

p  p 

then  the  pressure  of  the  mixture  will  be  p',  according  to  the 
preceding  experimental  fact  (Art.  56).  Now  if  the  mixture 
be  compressed  till  its  volume  is  v,  its  temperature  remain- 
ing constant,  the  pressure  will  become,  by  Mariotte's  law, 

P  +/• 

This  result  is  equally  true  for  a  mixture  of  any  number 
of  gases. 


108  VAPORS,    GASES. 

58.  Mixture  of  Unequal  Volumes  of  Gases  hav- 
ing Unequal  Pressures.— Tm;o  volumes  v,  v',  of  dif- 
ferent gases,  at  the  respective  pressures  p,  p' ,  are  mixed 
together  so  that  the  volume  of  the  mixture  is  V;  to 
find  the  pressure  of  the  rnixture. 

Change  the  volume  of  each  gas  to  V;  their  pressures  will 
be,  respectively  (Art.  48), 

V  v'    , 

and  therefore  (Art.  57)  the  pressure  of  the  mixture  is 
V       ,    v'    , 

yP+yP  I 

and  if  P  be  this  pressure,  we  have 

PV  =  pv  +  p'v'. 
(See  Besant's  Hydromechanics,  p.  114.) 

59.  Vapors,  Gases.— The  term  vapor  is  applied  to 
those  gaseous  bodies,  such  as  steam,  which  can  be  liquefied 
at  ordinary  pressures  and  temperatures  ;  while  the  word  gas 
generally  denotes  a  body  which,  under  ordinary  conditions, 
is  never  found  in  any  state  but  the  gaseous.  The  laws 
already  stated  of  gases  are  equally  true  of  vapors  within 
certain  ranges  of  temperature,  the  only  difference  between 
the  mechanical  qualities  of  vapors  and  gases,  as  distinguished 
from  their  chemical  qualities,  being  that  the  former  are 
easily  condensed  into  liquids  by  lowering  the  temperature, 
while  the  latter  can  be  condensed  only  by  the  application 
either  of  great  pressure  or  extreme  cold,  or  a  combination  of 
both. 

Prof.  Faraday  succeeded  in  condensing  a  number  of  different  gases  ; 
he  found  that  carbonic  acid,  at  the  temperature  of  —11",  was  liquefied 
by  a  pressure  of  20  atmospheres,  but  when  it  was  at  the  temperature 


FORMATION  OF   VAPOR,    SATURATION.  109 

of  0",  a  pressure  of  36  atmospheres*  was  required  to  produce  conden- 
sation. 

In  1877,  M.  Pictet  succeeded  in  liquefying  oxygen  by  subjecting  it 
to  a  pressure  of  300  atmospheres ;  at  the  close  of  the  same  year,  M. 
Cailletet  effected  the  liquefaction  of  nitrogen,  hydrogen,  and  atmos- 
pheric air.  Such  experimental  results  point  to  the  general  conclusion 
that  all  gases  are  the  vapors  of  liquids  of  different  kinds,  f 

60.  Formation  of  Vapor,  Saturation. — The  major- 
ity of  liquids,  when  left  to  themselves  in  contact  with  the 
atmosphere,  gradually  pass  into  the  state  of  vapor  and  dis- 
appear. Tliis  phenomenon  occurs  much  more  rapidly  with 
some  liquids  than  with  others.  Thus,  a  drop  of  ether  dis- 
appears almost  instantaneously ;  alcohol  also  evaporates  very 
quickly;  but  water  evaporates  much  more  slowly.  If  water 
be  introduced  into  a  space  containing  dry  air,  vapor  is  im- 
mediately formed  ;  if  the  temperature  be  increased,  or  the 
space  enlarged,  the  quantity  of  vapor  will  be  increased  ;  but 
if  the  temperature  be  lowered,  or  the  space  diminished,  some 
portion  of  the  vapor  will  be  condensed;  in  all  cases  the 
pressure  of  the  air  will  be  increased  by  the  pressure  due  to 
the  vapor  thus  formed.  The  formation  of  vapor  is  inde- 
pendent of  the  presence  of  air  or  of  its  density,  the  only 
effect  which  the  air  produces  being  a  retardation  of  the 
time  in  which  the  vapor  is  formed.  If  water  be  introduced 
into  a  vacuum,  it  is  iustantaneously  filled  with  vapor,  but 
the  quantity  of  vapor  is  the  same  as  if  the  space  had  been 
originally  filled  with  air. 

While  the  supply  of  water  remains,  as  a  source  from 
which  vapor  can  be  produced,  any  given  space  will  be 
always  saturated  with  vapor,  *.  e.,  there  will  be  as  much 
Yapor  as  the  temperature  admits  of.  If  the  temperature  be 
lowered,  a  portion  of  the  vapor  will  be  immediately  con- 
densed, and  become  visible  in  the  form  of  a  liquid ;  but  if 

*  An  atmosphere  denotes  the  pressure  due  to  a  column  of  mercury  29.9  inches  in 
height. 

t  Besant's  Hydrostatics,  p.  136. 


110  CHANGE   OF   VOLUME  AND    TEMPERATURE. 

tlie  temperature  be  increased  so  that  all  the  water  is  turned 
into  vapor,  then  for  this  and  all  higher  temperatures  the 
pressure  of  the  vapor  will  change  in  accordance  with  the 
same  law  which  regulates  the  connection  between  the  press- 
ure and  temperature  of  gases  (Art.  53). 

The  atmosphere  always  contains  more  or  less  aqueous 
vapor,  and  if  p  be  th6  pressure  of  dry  air,  and  tt  of  the 
vapor  in  the  atmosphere  at  any  time,  the  actual  pressure  of 
the  atmosphere  is  p-{-'rT. 

61.  Volume  of  Atmospheric  Air  without  its  Ta- 
por. — Having  given  the  pressures  of  a  volume  v  of 
atmospherie  air,  and  of  the  vapor  it  contains,  to  find 
the  volume  of  the  air  ivithout  its  vapor  at  the  same 
pressure  and  temperature. 

Let  P  be  the  pressure  of  the  atmosphere  and  p  that  of 
the  vapor;  and  let  v'  be  the  required  volume  of  the  air 
without  its  vapor,  at  the  pressure  P.  Then  P — ^  is  the 
pressure  of  the  air  alone  when  its  volume  is  v.  Hence  we 
have  (Art.  48), 

P  :  P—p  ::  v  :  v'  -, 

P  —  p 


62.  Pressure  of  Gas  when  Volume  and  Temper- 
ature are  Clianged. — J.  gas  contained  in  a  closed 
vessel  of  volume  v  is  in  contact  with  water,  and  its 
pressure  at  the  temperature  t  is  P ;  it  is  required  to 
determine  its  pressure  when  v  is  changed  to  v'  and  t 
to  t'. 

Let  p  and  p'  be  the  pressures  of  the  vapor  at  the  temper- 
atures t  and  t',  respectively,  and  P'  the  required  pressure. 

Then  P — p  and  P'  — p'  are  the  pressures  of  the  gas 
alone,  under  the  two  sets   of  conditions  stated.     Hence, 


EXAMPLE.  Ill 

calling  p  and  p  the  densities  of  the  gas,  we  have,  from  (3) 
of  Art.  54, 

P-p  =  kp{l  +  at), 

P'  -  p'  =  kp'  (1  +  at') ; 
also,  from  (1)  of  Art.  48,  we  have  vp  =  v'p'. 
P'  —p'        V  1  +  at' 


"      P-p         v'l  +  af 
which  gives  the  value  of  P'. 


(1) 


Cor. — If  a  and  a'  be  the  densities  of  vapor  under  the  two 
conditions,  we  have 

p    ~    o{l  +  at)'  ^^ 

Dividing  (1)  by  (2),  we  get 

p  P  — p'  va  _ 

~p'  T'^   ~  rV ' 

-  =  §^^!-  (3) 

va         P  p  —  pp 

If  Pp'  >  P'p,  v'o'  will  exceed  va ;  i.  e.,  more  vapor  will 
have  been  absorbed  by  the  gas.  But  if  Pp'  <  P'p,  then 
v'a'  will  be  less  than  va,  and  the  gas  must  therefore,  in 
changing  its  volume  and  temperature,  have  lost  a  portion 
of  its  vapor.     (See  Besant's  Hydrostatics,  p.  138.) 

EXAMPLE. 

Having  given  the  pressures  P  and  p  of  a  volume  v  of 

atmospheric  air,  and  of  the  vapor  it  contains,  to  find  the 

volume  of  the  air,  without  its  vapor,  at  the  same  pressnre 

P,  the  temperature  remaining  constant. 

P  —  p 
Ans.  Volume  of  an-  =  — 75—  y. 


113  PRESSURE   OF  VAPOR  IN  THE  AIR. 

63.  Formatiou  of  Dew,  the  Dew  Point. — Dew  is 

the  name  given  to  those  drops  of  water  which  are  seen  in 
the  morning  on  the  leaves  of  plants,  and  are  especially 
noticeable  in  the  spring  and  antumn.  If  any  portion  of  the 
space  occupied  by  the  atmosphere  be  saturated  with  vapor, 
i.  e.,  if  the  density  of  the  vapor  be  as  great  as  it  can  be  for 
the  temperature,  then  the  slightest  fall  of  temperature  will 
produce  condensation  of  some  portion  of  the  vapor  ;  but  if 
the  density  of  the  vapor  be  not  at  its  maximum  for  that 
temperature,  no  condensation  will  take  place  until  the  tem- 
perature is  lowered  below  the  point  corresponding  to  the 
saturation  of  the  space. 

If  any  body  in  contact  with  the  atmosphere  be  cooled 
down  until  its  temperature  is  below  that  w^hich  corresponds 
to  the  saturation  of  the  air  around  it,  condensation  of  the 
vapor  will  take  place,  and  the  condensed  vapor  will  be 
deposited  in  the  form  of  deto  upon  the  surface  of  the  body. 
Heat  radiates  from  the  ground,  and  from  the  bodies  upon 
it,  and  unless  there  are  clouds  from  which  the  heat  would 
be  radiated  back,  the  surfaces  are  cooled,  and  the  vapor  in 
the  adjacent  stratum  of  the  atmosphere  condenses  and  falls 
in  small  drops  of  water  on  the  surface.  The  formation  of  dew 
on  the  ground  depends  therefore  on  the  cooling  of  its  surface, 
and  this  is  in  general  greater  and  more  quickly  effected  when 
the  sky  is  free  from  clouds.  This  accounts  for  the  dew 
with  which  the  ground  is  covered  after  a  clear  night.  A 
covering  of  any  kind  will  diminish  the  formation  of  dew 
beneath ;  for  instance,  but  very  little  dew  will  be  formed 
under  the  shade  of  large  trees. 

The  dew-point  is  the  temperature  at  which  vapor  begins 
to  be  deposited  in  the  form  of  dew,  and  it  must  be  deter- 
mined by  actual  observation. 

64.  Pressure  of  Yapor  in  the  Air. — Tables*  have 

*  Besant's  Hydrostatics,  p.  143. 


MAXIMUM  DENSITY  OF   WATER.  113 

been  formed  and  empirical  formulas  constructed  for  deter- 
mining the  relation  between  the  temperature  and  the  elastic 
force  of  vapor,  at  the  saturating  density,  for  certain  ranges 
of  temperature.  If,  therefore,  the  dew-point  be  ascertained, 
we  can  at  once  determine  the  pi'essure  of  the  vapor  in  the 
air  by  means  of  these  tables.  For,  if  t'  be  the  dew-point, 
and  p'  the  corresponding  pressure,  then  at  any  other  tem- 
perature t  of  the  air  above  t',  we  have,  for  the  required 
pressure, 

65.  Eflfect  of  Compression  or  Dilatation  on  the 
Temperature  of  a  Gas.— It  is  an  experimental  fact  that, 
if  a  quantity  of  air  be  suddenly  compressed,  its  temperature 
is  raised ;  and  that,  if  the  compression  be  of  small  amount, 
the  relative  increase  of  temperature  is  proportional  to  the 
condensation.  Thus,  if  the  density  be  changed  from  p  to 
p',  the  increase  of  temperature  is  proportional  to 

P'  -  P 


If  the  air  be  allowed  to  dilate,  its  temperature  is  dimin- 
ished according  to  the  same  law.  A  stream  of  compressed 
air  when  issuing  from  a  closed  vessel  is  sensibly  chilled.  The 
reason  that  the  compression  or  dilatation  must  be  sudden, 
is  that  no  heat  should  be  allowed  to  escape,  or  to  be  admit- 
ted. If  the  experiment  be  performed  in  a  non-conducting 
vessel,  there  is  no  necessity  for  rapidity  of  action. 

66.  Expansion  of  Bodies  —  Maximum  Density  of 
Water. — In  general,  all  solid  and  liquid  bodies  expand 
under  the  action  of  heat,  and  contract  when  heat  is  with- 
drawn. The  expansion  of  mercury  is  proportional  to  the 
increase  of  temperature,  within  certain  limits;  this  is  also 
the  case  with  solid  bodies,  such  as  glass  and  steel.    For 


114  THERMAL    CAPACITTSPECIFW  HEAT. 

water  and  aqueous  bodies  generally,  the  law  of  expansion  is 
unknown. 

It  is  a  remarkable  property  of  water  that,  at  a  tempera- 
ture of  about  4°  C.  or  40°  F.,  its  volume  is  a  minimum  and 
therefore  its  density  is  a  maximum ;  *  and  whether  its  tem- 
perature increases  or  decreases  from  this  point,  the  water 
expands  in  volume.  When  the  temperature  descends  to  the 
freezing  point,  there  is  a  still  further  expansion  at  the 
moment  of  congelation;  for  this  reason,  ice  floats  in 
water. 

We  can  now  see  what  takes  place  in  a  pond  of  fresh 
water  during  winter.  The  fall  of  temperature  at  the  sur- 
face of  the  pond  does  not  extend  to  the  bottom,  where  the 
water  seldom  falls  below  4°  C,  whatever  may  be  the  exter- 
nal temperature.  As  the  temperature  at  the  surftice  de- 
scends, the  water  at  the  surface  cools,  and  being  conti-acted, 
it  becomes  heavier  than  the  water  beneath,  and  sinks  to  the 
bottom.  The  water  from  beneath  rises  and  becomes  cooled 
in  its  turn;  and  this  process  goes  on  till  all  the  water  has 
attained  its  maximum  density,  i.e.,  till  its  temperature  is 
4°  C.  But  when  all  the  water  has  attained  this  tempera- 
ture, it  will  remain  stationary  ;  and  any  further  cooling  of 
the  water  at  the  surface  will  expand  it,  until  it  finally  con- 
geals. It  is  clear  that  the  deeper  the  water  is,  the  longer 
will  be  the  time  before  the  whole  of  the  water  has  attained 
its  maximum  density,  and  therefore  that  ice  will  form  much 
less  rapidly  on  the  surface  of  deep  than  on  the  surface  of 
shallow  ponds. 

It  is  from  the  fact  that  water  expands  in  freezing,  taken 
in  connection  with  the  low  conducting  power  of  liquids 
generally,  that  the  temperature  at  the  bottom  of  deep  ponds 
remains  moderate  even  during  very  severe  cold,  and  that  the 
lives  of  aquatic  animals  are  preserved. 

*  The  results  of  Playfair  and  Joule  givo  3°.945C.  as  the  temperature  at  which  the 
density  is  a  maximum.    Phil.  Transactions,  1856. 


THERMAL    CAPACITT^SPECIFIC  HEAT.  115 

67.  Thermal  Capacity  —  Uuit  of  Heat  —  Specilic 
Heat. — The  thermal  capacity  of  a  body  is  the  quantity  of 
heat  required  to  raise  the  temperature  of  the  body  one 
degree. 

The  unit  of  heat  wliich  is  generally  employed  is  the  quan- 
tity of  heat  required  to  raise  a  unit  mass  of  water  through 
one  degree  C,  the  temperature  of  the  water  being  between 
0°  C.  and  40°  C.    It  is  called  tlie  thermal  unit  Centigrade. 

The  specific  heat  of  a  body  is  the  thermal  capacity  of  a 
unit  of  its  mass  ;  and  it  is  always  to  be  understood  that  the 
same  unit  of  mass  is  employed  for  the  body  as  for  the  water 
mentioned  in  the  definition  of  the  unit  of  heat.  Therefore, 
specific  heat  is  independent  of  the  unit,  and  is  merely  the 
ratio  of  the  quantity  of  heat  required  to  increase  by  1^  the 
temperature  of  the  body  to  the  quantity  of  heat  required  to 
increase  by  1°  the  temperature  of  an  equal  mass  of  water. 

The  quantity  of  heat  expended  in  changing  the  tempera- 
ture from  t  to  t' 

varies  as  t'  —  t  when  the  mass  is  given, 

and  varies  as  the  mass  when  t'  —  t  is  given  ; 

and  therefore  generally  it  varies  as  m  {f  —  t),  if  m  be  the 
mass.  Hence,  the  quantity  of  heat  expended  in  changing 
the  temperature  of  the  mass  m  from  t  to  t'  is 

sm  {f  —  t),  (1) 

where  s  is  the  specific  heat  of  the  substance,  since  it  is  the 
quantity  of  heat  required  to  raise  by  1"  the  temperature  of 
the  unit  of  mass,  which  may  be  shown  by  putting  m  =  1 
and  f  —  t  =  1. 

Let  dH  denote  the  quantity  of  heat  which  produces  in 
the  unit  of  mass  a  change  of  temperature  dt,  then  the  meas- 
ure of  the  specific  heat  is   -jT- 


116  SPECIFIC  HEAT  AT  CONSTANT  PRESSURE. 

68.  Comparison  of  Specific  Heat  at  a  Constant 
Pressure  with  that  at  a  Constant  Yolume.— In  the 

specific  heat  of  gases  there  are  two  cases  to  be  considered  : 
(1)  when  the  pressure  remains  constant,  the  gas  being 
allowed  to  expand  ;  (2)  when  the  volume  is  constant. 

Let  the  pressure  p  remain  constant  while  the  application 
of  a  small  quantity  of  heat  H  increases  the  temperature  T 
by  T,  and  changes  the  density  from  p  to  p'.  From  (3)  of 
Art.  55,  by  putting  ha  ■=  K^  Ave  have 

p  =  KpT=Kp'{T+T).  (1) 

Now  if  the  air  be  rapidly  compressed  into  its  original 
volume,  its  temperature  will  be  increased  (Art.  65),  and  we 
shall  have 

the  increase  of  temperature  p  —  p' 

^      =  /^  J  [by  (1)], 

where  ju  is  a  constant. 

.'.     the  increase  of  temperature  =  jur,  (3) 

and  hence  the  whole  change  of  temperature  produced  by 
the  heat  H,  when  the  volume  is  constant, 

=  r  +  /i-  =  At.  (3) 

In  order,  therefore,  to  produce  a  change  of  temperature 
T  when  the  volume  is  constant,  the  quantity  of  heat  required 

TT 

is  -r-,  and  consequently, 

specific  heat  at  constant  pressure  H  .  . 

specific  heat  at  constant  volume  H 

1 

Cor. — Therefore  the  specific  heat  at  a  constant  pressure 
exceeds  the  specific  heat  at  a  constant  volume;  and  this 


EXAMPLES.  117 

excess  from  (3)  is  equal  to  the  quantity  of  heat  ut  that  is 
disengaged  when  the  gas  is  suddenly  compressed  into  its 
original  volume. 

ScH. — The  value  of  a  is  found  experimentally  to  be  con- 
stant for  all  simple  gases,  its  value  being  approximately 
1.408.     (See  Besant's  Hydromechanics,  p.  118.) 

EXAMPLES. 

1.  A  mass  m^  of  a  substance  of  specific  heat  s^  and  tem- 
perature t^,  is  mixed  with  a  mass  Wg  of  a  substance  of  spe- 
cific heat  Sg  and  temperature  t^,  the  mixture  being  merely 
mechanical,  so  that  no  heat  is  generated  or  absorbed  by  any 
action  between  the  substances,  and  all  gain  or  loss  of  heat 
from  external  sources  is  prevented.  Find  the  resulting 
temperature  /  of  the  mixture. 

Suppose  the  former  body  to  be  the  warmer ;  then  it  cools 
down  from  t^  to  t,  while  the  colder  rises  from  i^g  to  t. 
Therefore  we  shall  have 

m^s^  (/j  —  0  =  t'i6  units  of  heat  lost  by  the 
former  body, 

and  /Wg^g  {t  —  /g)  =  the  units  of  heat  gained  by  the 

latter  body, 

and  since  the  quantity  of  heat  lost  by  the  warmer  body  is 
equal  to  that  gained  by  the  cooler,  these  two  expressions 
are  equal ;  therefore 

*Wi«i  (^1  —  0  =  '"^z^i  (^  —  h)  5 

One  of  the  methods  of  finding  the  specific  heat  of  a  sub- 
stance is  by  immersing  it  in  a  given  weight  of  water,  and 
observing  the  temperature  attained  by  the  two  substances. 


118  SUDDEN  COMPRESSION  OF  A   MASS   OF  AIR. 

2.  A  mass  Jf  of  a  substance  of  specific  heat  S  and  tem- 
perature T,  is  immersed  in  a  vessel  of  water,  m'  and  m 
being  the  masses  of  the  vessel  and  of  the  water  in  it,  and 
t'  their  common  temperature  and  s'  the  specific  heat  of  the 
vessel.  Find  the  temperature  t  of  the  whole  after  immer- 
sion. .  _  MST  -{-  lilt'  +  m's't' 

MS  +  w  +  m's' 

3.  A  glass  vessel  weighing  1  lb.  contains  5  oz.  of  water, 
both  at  20°,  and  2  oz.  of  iron  at  100°  is  immersed.  What 
is  the  temperature  of  the  whole,  taking  .2  as  the  specific 
heat  of  glass  and  .12  of  iron  ?  Ans.  22°^^. 

The  following  are  approximate  values  of  the  specific  heats 
of  a  few  substances : 

Water, 1 

Thermometer-glass,    ....  0.198 

Iron, 0.114 

Zinc, 0.1 

Mercury, 0.03 

Silver, 0.06 

Brass, 0.09 

(Besant's  Hydrostatics,  p.  147.) 

69.  Sudden  Compression  of  a  Mass  of  k\v,—A 
mass  of  air  being  suddenly  *  compressed  or  dilated,  it 
is  required  to  find  the  new  pressure  and  temperature. 

Let  p,  p,  T  be  the  pressure,  density,  and  absolute  tem- 
perature at  any  stage  of  the  process ;  p,  p',  T'  the  new 
pressure,  density,  and  temperature ;  and  let  dThe  the  change 
of  temperature  due  to  the  change  dp  in  p.    Then  we  have 

dT         dp  ,^- 

*  If  the  compression  takes  place  in  a  non-conducting  vessel,  so  tbat  no  heat  is 
lost  or  gained,  the  compression  need  not  be  rapid. 


SUDDEN  COMPRESSION  OF  A   MASS    OF  AIR.         119 

From  (1)  of  Art.  68,  we  have 

p  =  KpT',  (2) 

=  KT  +  KuT  [from  (1)].  (3) 

Dividing  (3)  by  (2),  we  have 

^^  =  ^+^  =  ^  [from  (3)  of  Art.  68], 
p  dp       p       p       p   ^  ^  '  -■ 

dp       Xdp  ... 

or,  -^  =  — ^.  (4) 

P  P 

Integrating  between  the  limits  p'  and  p,  p'  and  p,  we  have 
p[  _  IP\\ 

p  ~  \pr 

which  determines  the  pressure. 

Also,  p'  =  Kp'T', 

which,  divided  by  (2),  gives  * 


p         pT 


(6) 


From  (5)  and  (6),  we  have 

^P 


T  ~\pl 


r  =  Tipj]  (7) 


which  determines  the  temperature.     (See  Besant's  hydro- 
mechanics, p.  118.) 


120      HEIGHT  OF  THE  HOMOGENEOUS  ATMOSPHERE. 

70.  Mass  of  the  Earth's  Atmosphere. — By  means 
of  the  barometer,  some  idea  may  be  formed  of  the  mass  of 
air  and  vapor  surrounding  the  earth,  since  the  weight  of  the 
whole  atmosphere  is  equal  to  that  of  a  stratum  of  mercury 
about  29.9  inches  thick  covering  the  globe.  Suppose  the 
earth  to  be  a  sphere  of  radius  r,  and  that  h  is  the  height  of 
the  barometric  column  at  all  points  of  its  surface.  Then 
the  mass  of  the  atmosphere  is  approximately  equivalent  to 
the  mass  4Trcrr%  of  mercury,  where  a  is  the  density  of  the 
mercury. 

Let  p  be  the  mean  density  of  the  earth  ;  then, 
the  mass  of  the  atmosphere  :  the  mass  of  the  earth 

=:  4iTrar%  :  f  ^p?"^ 
=  3a/i  :  pr. 

Taking  a  =  13.568  (Art.  47),  and  p  =  5.5,*  and  sup- 
posing the  height  of  the  barometric  column  h  to  be  30 
inches,  which  is  probably  near  the  average  height  at  sea- 
level,!  it  will  be  found  that  the  above  ratio  of  the  mass  of 
the  atmosphere  to  that  of  the  earth  is  about  xi^ieoir- 

71.  The  Height  of  the  Homogeneous  Atmosphere. 

— If  the  atmosphere  were  of  the  same  density  throughout 
as  at  the  surface  of  the  earth,  its  height  I  would  be  approx- 
imately obtained  from  the  following  equation, 

ah  =  pi,  (1) 

where  a  and  p  are  the  densities  of  mercury  and  air  respect- 
ively, and  h  is  the  height  of  the  barometric  column.  From 
Art.  70,  and  Art.  33,  Sch.,  we  have 

a  =  13.568  x768p  —  10420.224p, 

*  There  is  gome  doubt  about  the  accuracy  of  this  value  ;  the  value  deduced  by 
the  Astronomer  Royal  at  the  Harton  Colliery  in  1854  is  6.6.    Phil.  Trans.,  1856. 
t  See  Bncy .  Brit. ,  Vol  III. ,  p.  28. 


LIMIT  TO    THE  HEIGHT  OF  THE  ATMOSPHERE.      121 

aud  taking  h  =  30  inches,  we  have,  by  solving  (1)  for  I, 

I  =  h-  =  26050  feet, 
9 

which  is  a  little  less  than  5  miles. 


TZ.  Necessary  Limit  to  the  Height  of  the  At- 
mospliere. — Since  the  attraction  of  the  earth  diminishes 
at  a  distance  from  its  surface  (Anal.  Mechs.,  Art.  133«f),  it 
is  clear  that  the  atmosphere  is  very  far  from  being  of  uni- 
form density  throughout,  and  tlierefore  the  result  in  Art.  71 
is  very  far  from  the  truth.  A  limit  can  be  found,  however, 
to  the  height  of  the  atmosphere  from  the  consideration  that, 
beyond  a  certain  distance  from  the  earth's  centre,  its  attrac- 
tion will  be  unable  to  retain  the  particles  of  air  in  the  cir- 
cular paths  which  they  describe  about  the  earth,  since  the 
centrifugal  force  must  exceed  the  force  of  gravity. 

Let  cj  be  the  earth's  angular  velocity,  and  r  its  radius. 
Then  the  centrifugal  force  of  a  particle  tn  of  air  on  the 

earth's   surface  is   wiwV,  and   this  is  equal  to  -r^^  [Anal. 

2o9 

Mechs.,  Art.  199,  (3)] ;   therefore,  at  a  height  z  above  the 

surface,  the  centrifugal  force  mw^  (^  -[.  £) 

_    mg  r  -\-  z 
"^  289  ~1' 

The  earth's  attraction  at  the  same  height  (Anal.  Mechs., 
Art.  133«) 

_     mgr^ 

-  (TT^p' 

and,  in  order  that  the  particle  may  be  retained  in  its  path, 
these  two  forces  must  equal  each  other. 

mg  r  +  z mgr^ 

•'■    289  ~7~  "  (T+'^a' 


122      DECREASE   OF  DEN  SIT  r  OF  THE  ATMOSPHERE. 

r  -\-  z '/'^ 

®^'  "2897  ~  {Th^f 

.:    z  =  r(V289  — l); 
=  5.6r  + 
=  22000  miles  (approximately). 

Rem. — The  actual  height  of  the  atmosphere,  however,  is 
possibly  much  lower  than  this,  for  its  temperature  has  been 
found,  by  experiments  made  in  balloons,  to  diminish  with 
great  rapidity  during  an  ascent;  it  is  therefore  very  likely 
that,  at  a  height  less  than  5r,  the  air  may  be  liquefied  by 
extreme  cold,  and  in  that  case  its  external  surface  would  be 
of  the  same  kind  as  the  surfaces  of  known  inelastic  fluids. 
(Besant's  Hydromechanics,  p.  120.) 

73.   Decrease   of  Density   of  the  Atmosphere. — 

(1)    ~[Vlien  the  force  of  gravity  is  constant. 

Take  a  vertical  column  of  the  atmosphere,  and  let  it  be 
divided  into  an  indefinite  number  of  horizontal  strata  of 
equal  thickness,  so  that  the  density  of  the  air  may  be  uni- 
form throughout  the  same  stratum.  Let  the  weight  of  the 
whole  column  from  the  top  of  the  atmosphere  to  the  earth 
=  a,  that  of  the  whole  column  above  the  lowest  stratum  = 
h,  that  of  the  column  above  the  second  =:  c,  and  so  on. 
Then  h,  c,  d,  etc.,  are  the  forces  respectively  which  compress 
the  first,  second,  third,  etc.  strata,  which,  as  they  are  of 
equal  thickness,  are  as  their  weights,  a  —  l,  h  —  c,  c  —  d, 
etc.     Hence  we  have 

a  —  h  '.  h  —  c  :'.  h  :  c, 

.'.     a  '.  h  ::  h  :  c. 

In  the  same  way,  it  may  be  shown  that 

h  :  c  ::  c  '.  d, 

and  so  on.     Hence,  h,  c,  d,  etc.,  and  therefore  the  densities 


DECREASE   OF  DENSITY  OF  THE  ATMOSPHERE.       123 

of  the  successive  strata,  form  a  series  of  terms  in  geometric 
progression,  which  is  decreasing  since  a  is  greater  than  h, 
and  therefore  h  greater  than  c,  and  so  on  ;  and  as  the  strata 
all  have  the  same  thickness,  the  heights  of  the  several  strata 
ahove  the  earth's  surface  increase  in  arithmetic  progression. 
Hence, 

//  a  seHes  of  heights  be  taken  in  arithmetic  pro- 
gression, when  the  force  of  gravity  is  constant,  the 
densities  of  the  air  decrease  in  geometric  progression. 

ScH. — By  barometric  observations  at  different  altitudes, 
it  is  found  that  at  the  height  of  3|-  miles  above  the  earth's 
surface,  the  air  is  about  one-half  as  dense  as  it  is  at  the 
surface.  Forming  therefore  an  arithmetic  series,  with  Z^ 
for  the  common  difference,  to  denote  the  heights,  and  a 
geometric  series  with  \  for  the  common  ratio,  to  denote 
densities,  we  have 

Heights,  3|,  7,  10^,  14,  17|,  21,  U\,  28,  31|,  35,  etc. 
Densities,    \,  \,    \,    -^,    -^,  -^^,  y^,  ^-g-,  -^,  -r^^,etc. 

That  is,  according  to  this  law,  at  the  height  of  35  miles 
the  air  is  less  than  a  thousandth  part  as  dense  as  it  is  at  the 
surface  of  the  earth. 

(2)  When  the  force  of  gravity  varies  inversely  as  the 
square  of  the  distance  from  the  earth's  centre. 

Let  r  be  the  radius  of  the  earth,  p'  the  density  at  the  sur- 
face of  the  earth,  p  the  density  at  a  height  z,  and  h  the 
height  of  a  homogeneous  atmosphere.  Then,  since  the 
density  varies  as  the  compressing  force,  and  this  varies  as 
the  weight,  we  have 

p'  :  dp  ::  hp'g  :  g -^—^-{— p  dz), 

where  a  and  -—- — -.  are  the  measures  of  the  earth's  attrac- 

•  {r  -\-  z)^ 


124        HEIGHTS  DETERMINED   BY  THE  BAROMETER. 

tion  at  the  surface  and  at  a  height  z,  the  negative  sign  being 
taken  because  the  density  is-  a  decreasing  function  of  the 

height  z. 

dp  _       r^      dz 
p  h  {r  -{-  z)^ 

Integrating,   observing  that  when    z  =  0,    p  =  p',    we 
have 


,      p        r^(     1  1\ 


p'       h  \r  + 

P' 


P  = 


A  \r  ""  r+ej 


which  shows  that,  ii  r  -{-  z  increases  in  liarnionic  progres- 
sion,   will  decrease   in  arithmetic  progression,  and 

therefore  p  will  decrease  in  geometric  progression.    Hence, 

//  a  series  of  heights  be  taken  in  harmonic  progres- 
sion, when  the  force  of  gravity  is  regarded  as  variable, 
the  densities  of  the  air  decrease  in  geometHc  progres- 
sion.    (See  Bland's  Hydrostatics,  p.  258.) 

74.  Heights  Determined  l)y  the  Barometer. — A 

very  important  use  of  the  barometer  is  to  find  the  difference 
of  level  of  two  places  situated  at  unequal  distances  above 
the  surface  of  the  earth.  Since  the  height  of  the  column  of 
mercury  in  the  barometer  depends  on  the  pressure  of  the 
atmosphere  (Art.  43),' and  as  the  pressure  of  the  atmosphere 
at  any  point  depends  upon  the  height  of  the  column  of  air 
extending  from  that  point  to  the  top  of  the  atmosphere,  it 
follows  that  this  pressure  will  decrease  as  we  ascend  above 
the  earth's  surface,  and  therefore  that  the  height  of  the 
column  of  mercury  will  diminish.  Tliat  is,  the  mercury  in 
the  barometer  will  fall  when  the  instrument  is  carried  from 


HEIGHTS  DETERMINED  BY  THE  BAROMETER.       135 

the  foot  to  the  top  of  a  mountain,  and  will  rise  again  when 
it  is  returned  to  its  former  position. 

(1)  When  the  force  of  gravity  is  regarded  as  con- 
stant. 

Consider  a  vertical  column  of  the  atmosphere  at  rest 
under  the  action  of  gravity.  Let  z  be  taken  vertical  and 
positive  upwards ;  and  at  a  height  z,  let  p  be  the  pressure 
and  p  the  density.  The  pressure^,  at  any  height  z.  is  meas- 
ured by  the  weight  of  the  column  of  air  extending  from 
that  height  to  the  top  of  the  atmosphere;  and  the  element- 
ary pressure  dp  will  be  measured  by  the  weight  of  the  col- 
umn having  the  same  base  and  the  elementary  height  dz. 
Therefore,  if  A  be  the  area  of  the  section  of  the  column,  we 
have 

Adp  =  —  Agp  dz, 

or,  dp  =  —gp  dz,  (1) 

the  negative  sign  being  taken  because  the  pressure  /?  is  a 
decreasing  function  of  the  height  z. 

If  t  be  the  temperature,  we  have  from  (3)  of  Art.  54, 

p  =  Icp  (1  +  at).  (2) 

Dividing  (1)  by  (2),  we  have 

dp  _         gdz 

p    -~  T^at  ^^^ 

If  the  heights  above  the  earth's  surface  are  small,  the 
force  of  gravity  g  may  be  regarded  as  constant ;  and  sup- 
posing t  constant,  we  have,  by  inj;cgrating  (3), 

>^logA=^#=4>,  (-1) 

^  p  1  -{-  at   '  ^   ' 

where  J9'  is  the  pressure  at  the  height  z\ 


12G       HEIGHTS   DETERMINED   BY   THE  BAROMETER. 

Let  h,  h'  be  -the  observed  barometric  heights  at  the  two 
stations,  whose  altitudes  are  z  and  z' ;  let  a  be  the  density  of 
mercury  at  a  temperature  zero,  and  r,  r',  the  temperatures 
at  the  two  stations.     Then  we  have,  from  (2)  of  Art.  52, 

p  =  gah  (1  —  Bt), 

and  p'  =  gah'  (1  —  dr'), 

which  in  (4)  gives 

where  t  may  be  taken  approximately  equal  to  ^{t  -\-  r') ; 
from  this  equation  the  difference  of  the  heights  of  the  two 
stations  can  be  calculated. 

(2)  When  the  force  of  gravity  is  regarded  as  va- 
riable. 

If  the  heights  above  the  earth's  surface  be  considerable, 
it  is  necessary  to  take  account  of  the  variation  of  gravity  at 
lifferent  distances  from  the  earth's  centre. 

Calling  g  the  measure  of  the  earth's  attraction  at  the  level 
of  the  sea,  and  r  the  radius  of  the  earth,  then  we  have,  for 
the  measure  of  the  attraction  at  a  height  z, 

which,  being  substituted  in  (1)  for_«7,  gives 


dp  =  —  g  - — ; — -p  dz.  (7) 

^  ^  {r  -{■  zY^  ^  ' 

Dividing  (7)  by  (2),  we  have 

"  p   -        lJ^ai{r  +  zf  ^^' 


HEtGBTS  DETERMINED  BY  THE  BAROMETER.      127 

It  must  be  observed  that  p  is  the  sum  of  the  pressures 
due  to  the  air  itself,  and  to  the  aqueous  vapor  Avhich  is 
mixed  with  it ;  i.  e.,  the  quantity  kp  in  (2)  is  the  sum  of 
the  two,  kp,  k'p',  where  p  and  p  are  the  densities  of  the  air 
and  the  aqueous  vapor,  respectively. 

Considering  t  constant  as  before,  and  equal  to  the  mean 
of  the  temperatures  at  the  two  stations,  and  integrating  (8), 
we  have 

P        (1  +  «0  {r  +  z){r  +  z')  ^  ' 

As  before,  let  h,  h',  and  r,  t',  be  the  observed  barometric 
heights  and  temperatures,  and  a  the  density  of  mercury  at 
a  temperature  zero  ;  then  from  (2)  of  Art.  52,  by  substitut- 
ing for  g  its  value  from  (6),  we  have 

^       (r  -f  zf      ^  '* 

,  ^  -J^-  ah'  (1  -  Ot'), 

"     p   -\r  ^z'l  l-dr  k  ^^"^ 

Substituting  (10)  in  (9),  and  solving  for  z  —  z',  we  have 
z  —  z'  = 

gr^  \    ^h  ^r  +  z        ^i—drj 

(11) 
Since  6  is  very  small  (Art.  52),  we  have  . 

logl^'  =  log[l-e(T'-r)] 

=  -d{T'-r). 

(Calculus,  Art.  61.) 


128       HEIGHTS  DETERMIXED  BV  THE  BAROMETER. 

Substituting  this  in  (11),  and  reducing  N"aperian  to  com- 
mon logarithms  by  multiplying  by  m,  the  modulus  of  the 
common  svstem,  we  have 


,       k{\  +  at){r  +  z){r  +  z') 

z  —  z   = A 

mgr' 


log,o|+21og,o^, 
-md{T'-T)    ,     (12) 


from  which  the  value  of  z  can  be  determined  when  z'  is 
known. 

Cor.  1. — If  the  lower  station  be  nearly  at  the  level  of  the 
sea,  z'  =.  0,  and  (12)  becomes 

(13) 

Cor.  2. — In  the  above  investigation  no  account  has  been 

taken  of  the  variation  of  gravity  at  different  parts  of  the 

earth's  surface.    From  a  comparison  of  the  results  ol)taiiied 

by  causing  pendulums  to  oscillate  in  different  latitudes,  if  ^ 

be  the  measure  of  gravity  at  a  place  of  latitude  a,  and  g'  at 

a  place  of  latitude  A',  it  has  been  found  (Poisson,  Art.  628) 

that 

g_  _  1  —  .002588  cos  %X  ^ 

/  ~  1  —  .002588  cos  2/.' ' 

Tc  Ic   \  —  .002588  cos  2A' 

therefore,  —  = — ,-^ t.,.^-oo n^'  \\^) 

mg        mg  1  —  .002o88  cos  2A  ^     ' 

If  X'  be  the  latitude  of  Paris,  the  value  of  the  quantity 

_^   (1  _  .002588  cos  2A')  (15) 

mg  ^  ' 

is  nearly  18336  French  metres,*  or  about  60158.56  English 

*  A  French  metre  is  89.37079  inches. 


HEIGHTS  DETERMINED   BY  THE  BAROMETER.       129 

feet;  representing  this  numerical  quantity  by  c  and  substi- 
tuting it  in  (14),  we  get 

h  c 


z  = 


my       1  —  .002588  cos  2A' 
which  in  (13)  gives 

.(l  +  «0(l  +  '.)r        j^,  .      ^v 

=  1-002588  col2lL^"g'VT  +^  ^"^4^  +  ?) 

_^0(r'_T)^,     (16) 

from  which  the  yalue  of  z  may  be  determined  by  a  series  of 
approximations;  i.  e.,  an  approximate  value  must  be  first 

obtained  by  neglecting   -  ;    then   this  approximate   value 

must  be  substituted  for  z  in  --,  and  a  more  accurate  value 

will  be  obtained,  and  the  same  process  may  be  repeated,  if 
necessary.* 

ScH.  1. — When  -  is  very  small,  it  may  be  neglected  in 

(16).  It  has  been  found  in  practice,  however,  that  in  this 
case  the  results  are  more  accurate  by  employing  18,393 
metres  as  the  value  of  c.     (Duhamel,  p.  259.) 

In  order  that  the  heights  as  determined  by  the  barometer 
may  be  very  exact  in  practice,  certain  corrections  are  neces- 
sary. For  instance,  the  value  of  k  is  modified  by  the  fact 
that  the  density  of  aqueous  vapor  at  a  given  temperature 
and  pressure  is  less  than  the  density  of  dry  air  under  the 
same  circumstances ;  and  the  proportion  of  aqueous  vap.oi 
to  dry  air  will  generally  be  different  at  the  two  stations. 

♦  A  formula  for  this  is  given  in  Ency.  Brit.,  Vol.  m.,  p.  386,  involving  a  conpid- 
eration  of  densities  of  vapor. 


130 


SPECIFIC   GRAVITIES. 


ScH.  )l. — Formula  (16)  has  been  obtained  on  the  supposi- 
tion that  the  temperature  of  the  air  remains  constant  in 
passing  from  the  lower  to  the  higher  station ;  if,  however, 
the  difference  between  the  heights  be  very  great,  a  consid- 
erable error  may  be  thus  introduced,  and  formulse  have 
therefore  been  constructed  in  which  account  is  taken,  on 
various  hypotheses,  of  the  variation  of  atmospheric  temper- 
ature. A  formula  of  this  kind  is  given  in  Lindeman's 
Barometric  Tables,  constructed  on  the  supposition  that  the 
temperature  diminishes  in  harmonic  progression  through  a 
series  of  heights  increasing  in  arithmetic  progression. 

Also,  we  have  assumed  that  the  temperature  of  the 
mercury  in  the  barometer  is  the  same  as  that  of  the  air 
surrounding  it ;  but  in  some  cases,  as  for  instance  when  ob- 
servations are  made  in  a  balloon,  the  barometer  may  not 
remain  long  enough  in  the  same  place  to  acquire  the  tem- 
perature of  the  surrounding  air.  The  temperature  of  the 
mercury  may  be  observed,  however,  by  placing  the  bulb  of  a 
thermometer  in  the  cistern  of  the  barometer,  and  the  tem- 
peratures thus  obtained  must  be  used  in  (10).  (See  Be- 
gan t's  Hydromechanics,  p.  121.) 


SPECIFIC    GRAVITIES. 


Ratios  of  the  Specific   Gravities  of   different  sub- 
stances to  that  of  water  at  60°. 

Tin, 7.29 

Lead, 11.45 

Zinc, 6.86 

Nickel,    ....  8.38 

Iron, 7.844 

Flint-glass, .    .    .  2.5 

Marble,   .    .    .    .  2.716 

Rock-salt,    .    .    .  1.92 

Ivory,      .    .    .    .  1.917 


Diamond,     . 

.    .    3.52 

Sulphur, 

.    .    2 

Iodine,     .    . 

.    .    4.94 

Arsenic,  .    . 

.    .    5.96 

Gold,  .    -    . 

.    .  19.4 

Platina,  .    . 

.    .  21.53 

Silver, 

.    .  10.5 

Mercury, 

.    .  1.3.568 

Copper,  .    . 

.    .    8.85 

EXAMPLES. 


131 


Ice  (at  0°), 

Sea-water, 

Olive-oil, 


0.926 
1.027 
0.915 


Alcohol,.    .    .    .    0.794 
Ether,      .    .    .    .    0.724 


Ratios  of  the  densities  of  gases  and  vapors  of  differ- 
ent  substances  to  that  of  atmospheric  air  at  the  same 
temperature  and  under  the  same  pressure. 


Oxygen,  . 
Hydrogen, 
Nitrogen, 
Chlorine, 
Bromine, 
Iodine,     . 
Arsenic,  . 
Mercury, 


1.103 
0.069 
0.976 
2.44 
5.395 
8.701 
10.365 
6.978 


Water,     ....  0.62 

Alcohol,  .    .    .    .  1.613 

Carbonic  Acid,     .  1.524 

Ammonia,   .     .     .  0.591 

Sulphurous  Acid,  2.212 

Sulphuric  Acid,   .  2.763 

Ether,     .    .    .    .  2.586 


EXAMPLES. 

L  If  the  barometer  stand  at  28.372  inches,  find  the 
pressure  on  a  square  inch.  Ans.   13.902  lbs. 

2.  If  the  elastic  force  of  a  vapor  sustain  a  column  of 
mercury  3.34  inches  high,  find  its  pressure  on  a  square 
inch.  Ans.  1.64  lbs. 

3.  A  cubic  inch  of  mercury  at  16°  weighs  3429^  grs. 
nearly,  and  the  barometer  stands  at  30  inches.  Find  (1) 
the  atmospheric  pressure  on  the  square  inch  of  surface,  and 
(2)  the  height  of  a  barometer  filled  with  water  instead  of 
mercury,  the  specific  gravity  of  mercury  being  13.6. 

Ans.   (1)  14.698  lbs. ;  (2)  34  feet. 

4.  A  hollow  cylinder,  open  at  the  top,  is  inverted,  and 
partly  immersed  in  water.  It  is  required  to  find  the  depth 
of  the  surface  of  the  water  within  the  cylinder  below  the 
surface  of  the  water  without. 

Let  a  =  the  length  of  the  cylinder,  h  =  the  length  of 
the  part  not  immersed,  x  =  the  required  depth  of  the  sur- 


132  EXAMPLES. 

face  within  below  the  surfuco  without,  and  tt,  -',  the  press- 
ures of  the  atmospheric  air  and  of  the  compressed  air. 
Then  (Art.  48)  we  have 

•rr'  :  t:  ::  a  '.  h  +  x;  (1) 

also,  tt'  =  pressure  on  the  water  within  =  rr  +  gpx  =i 
gph  +  gpx,  if  h  be  the  height  of  the  water  barometer. 

Substituting  these  values  of  n-  and  rr'  in  (1),  we  have 

h  -\-  X  _      a 
h      ~  b  +  x' 

ViaA  +  {h  —  'bf  —  {h  +  b) 


X  = 


2 


5.  A  cylinder,  20  ft.  long,  is  half  filled  with  water,  and 
inverted  with  the  open  end  just  dipping  into  a  vessel  of 
water.  Find  the  altitude  of  the  water  in  the  cylinder,  the 
height  of  the  water  barometer  being  33  feet. 

A71S.  7.21  feet. 

6.  When  the  mercurial  barometer  stands  at  30  inches, 
what  is  the  height  of  the  barometer  formed  of  a  liquid 
whose  specific  gravity  is  5.6  ?       Ans.   72.7  inches  nearly. 

7.  The  air  contained  in  a  cubical  vessel,  the  edge  of 
which  is  one  foot,  is  compressed  into  a  cubical  yessel  of 
which  the  edge  is  one  inch.  Compare  the  pressures  on  a 
side  of  each  vessel.  Ans.  1  :  12. 

8.  If  the  elastic  force  of  a  mass  of  gas  whose  volume  is 
100  cubic  inches  be  30.275  inches  of  mercury,  find  its  elastic 
force  if  it  be  allowed  to  expand  to  a  volume  of  387  cubic 
inches.  Ans.   7.823  inches. 

9.  If  Fahrenheit's  Thermometer  mark  40°,  what  are  the 
corresponding  marks  of  Reaumur's  and  the  Centigrade  ? 

Ans.  3|;  4f,. 


EXAMPLES.  133 

10.  If  the  sum  of  the  readings  on  Fahrenheit's  and  the 
Centigrade  thermometer  be  zero  for  the  same  temperature, 
find  the  reading  of  each  thermometer. 

Ans.  llf  ;  —  llf 

11.  If  327  cubic  inches  of  gas  at  280°  be  allowed  to  cool 
down  to  56°,  find  the  volume.*       Ans.  223  cubic  inches. 

12.  If,  by  the  application  of  heat,  120  cubic  inches  at 
60°  F.  expand  into  180  cubic  inches,  find  the  temperature. 

Ans.  320°. 

13.  If  the  pressure  be  14.7  lbs.  on  the  square  inch  at  the 
temperature  62°,  what  will^t  become  if  raised  to  420°  ? 

Ans.  24.78  lbs. 

14.  If  the  pressure  at  50°  be  15  lbs.,  and  if  the  tempera- 
ture be  so  far  increased  as  to  make  the  pressure  21  lbs.,  find 
the  temperature.  Ans.  254°. 

15.  The  air  in  a  spherical  globe,  one  foot  in  diameter,  is 
compressed  into  another  globe,  6  inches  in  diameter,  and 
the  temperature  is  raised  t°.  Compare  (1)  the  pressures  of 
the  air  under  the  two  conditions,  and  (2)  tlie  pressures  on 
the  surfaces  of  the  globes.  j  (1)  1  :  8  (1  +  a^) ; 

^'^'•1(2)  l:2(l  +  «0. 

16.  The  temperature  of  the  air  in  an  extensible  spherical 
envelope  is  gradually  raised  t°,  and  the  envelope  is  allowed 
to  expand  till  its  radius  is  n  times  its  original  length.  Com- 
pare the  pressure  of  the  air  in  the  two  cases. 

Ans.  1  +  at  :  11^. 

17.  A  mass  of  air  at  a  temperature  t  is  contained  in  a 
cylinder  which  has  an  air-tight  piston  fitting  into  it,  and  it 
is  found  that  the  air  exerts  a  pressure  P  on  the  piston ;  the 

air  being  suddenly  compressed  into    -  of   its  former  .vol- 

*  Fahrenheit's  Thermometer  is  understood,  uiiless  otherwise  expressed. 


134  EXAMPLES. 

ume,  and  the  temperature  changed  to  ^^  find  the  presBore 
/"  on  the  piston.  .        p-  _  p   1  +  a^' 

***'         ~      **  1  +  at' 

18.  If  a  cubic  foot  of  gag,  whose  temperatnre  is  100**  and 
elastic  force  29^  inches,  be  cooled  down  to  40",  and  com- 
pressed by  a  force  equivalent  to  10^  inches,  find  its  Tolnme. 

Ans.  4334J7  cubic  inches. 

19.  If  20  cubic  inches  of  air,  whose  temperature  is  56° 
and  elastic  force  28.8  inches,  be  expanded  to  25  inches  by 
the  application  of  heat,  and  if  the  elastic  force  become  31 
inches,  find  the  temperature.  Antt.  234.27°. 

20.  Let  100  cubic  inches  of  air  have  a  temperatnre  32° 
and  a  pressure  29.922  inches;  if  the  temperature  become 
60°,  and  the  pressure  30  inches,  find  the  volume. 

An*.  105.42  cubic  inches. 

21.  A  cubic  foot  of  air  at  a  temperature  of  100°,  and 
under  a  pressure  of  29|^  inches  of  mercury,  is  cooled  down 
to  40°  and  compressed  by  an  additional  lOi  inches  of  mer- 
cury.    Find  the  volume.  Ans.  1137.86  cubic  inches. 

22.  If  h  and  //  be  the  heights  of  the  surface  of  the  mer- 
cury in  the  tube  of  a  barometer  above  the  surface  of  mercury 
in  the  cistern  at  two  different  times,  compare  the  densities 
of  the  air  at  those  times,  the  temperature  being  supposed 
unaltered,  Ans.  h  :  h'. 

23.  A  conical  wine-glass  is  immersed,  mouth  downwards, 
in  water.  How  far  must  it  be  depressed  in  order  that  the 
water  within  the  glass  may  rise  half-way  up  it  ? 

Ans.   th,  where  h  is  the  height  of  the  water  barometer. 

24.  A  cubic  foot  of  air  having  a  pressure  of  15  lbs.  on  a 
square  inch  is  mixed  with  a  cubic  inch  of  compressed  air, 
ha\ing  a  pressure  of  60  lbs.  on  a  square  inch.  Find  the 
pressure  of  the  mixture  when  it*  volume  is  1729  cubic 
inches.  Am.  15^41-^  lbs. 


EXAJn^LBS,  135 

39.  Twt>  TolomoSk  Fmd  JT,  of  different  gases^  a^  pr«>s&- 
jues  Pf  p\  and  temperatnie  4  sro  mixed  togetlia*;  the 
Tohime  of  the  mixlnro  k  U^  and  its  (empeanlme  I*.  Betep- 
mine  the  pressure^  J^F  +  yF^l  +  ^f 

26.  Three  gallons  of  water  at  45**  are  mixed  with  dx  gal- 
lons at  9(f .     What  is  the  temperature  of  the  mixture  ? 

An*.  79* 

27.  An  ounce  of  iron  at  120%  and  2  os.  of  line  at  90% 
are  thrown  into  6  oo.  of  water  at  10%  contained  in  a  glass 
vessel  weighing  10  oo.  What  is  the  final  temperature, 
taking  .1  and  J2  as  the  specific  heats  of  xinc  and  iron  ? 

Ams.  13®^. 


PART    II. 

HYDROKINETICS. 


CHAPTER    I. 


MOTION     OF     LIQUIDS.— EFFLUX.  — RESISTANCE    AND 
WORK    OF    LIQUIDS. 

75.  Telocity  of  a  Liquid  in  Pipes.—//  a  liquid 
run  through  any  pipe  of  vaHahle  diameter,  which  is 
kept  continualhj  full,  and  the  velocity  is  the  same  in 
every  part  of  a  transverse  section,  the  velocities  in  the 
different  transverse  sections  vary  inversely  as  the 
areas  of  the  sections. 

For  as  the  tube  is  kept  full,  and  the  liquid  is  incom- 
pressible (Art.  3),  the  same  quantity  of  liquid  which  runs 
through  one  section  will,  in  the  same  time,  run  through 
the  next  section,  and  so  on  through  any  other.  Hence  if 
k,  Tc'  be  the  areas  of  any  two  sections,  and  v,  v'  the  veloc- 
ities of  the  particles  at  those  sections,  we  have,  since  the 
quantity  of  liquid  which  flows  through  any  section  in  a 
unit  of  time  is  the  product  of  the  area  of  the  section  by  the 
velocity, 

lev  =  h'v' 'j 

.'.     V  :  v'   : :   k'  :  h  (1) 

Cor. — Hence,  as  the  section  of  a  mass  of  liquid  decreases, 
its  velocity  increases  in  the  same  proportion.  For  instance, 
the  velocity  of  a  stream  or  river  is  greater  at  places  where 
its  width  is  diminished.  This  demonstration  is  also 
applicable  to  different  sections  of  a  liquid  issuing  through 
the  orifice  of  a  vessel,  whether  the  section  be  taken  within 


VELOCITY  OF   EFFLVX.  137 

or  without  the  vessel,  provided  there  be  no  vficuity  in  the 
stream  between  the  sections. 

ScH. — It  is  supposed  in  this  proposition  that  the  changes 
in  the  diameters  of  the  sections  are  gradual,  and  nowhere 
abrupt ;  if  there  are  any  angles  in  the  pipe,  they  will 
produce  eddies  in  the  motion  of  the  liquid,  and  the  propo- 
sition will  not  hold  true. 

76.  Telocity  of  Efflux. — If  a  small  aperture  he 
made  in  a  vessel  containing  liquid,  the  velocity  with 
which  the  liquid  issues  from  the  vessel  is  the  same  as 
if  it  had  fallen  froin  the  level  of  the  surface  to  the 
level  of  the  aperture^ 

Let  EF  represent  a  very  small  orifice  in  the  bottom  of 

the  vessel  ABCD,  which  is  filled  with  a  liquid  to  the  level 

AB;   and  suppose  the  vessel  to  be 

kept  full  by  supplying  it  from  above, 

while    the    liquid    is    running    out 

through  the  orifice  EF.     Let  v  be 

the  velocity  of  efflux,  ^o  the  weight 

of  the  liquid  which  issues  with  that 

velocity  per  second,  and  h  the  height 

of    the    surface    above    the    orifice, 

called  the  head\  of  the  liquid.     Then  the  work  which  ic 

can    perform   while   descending  through   the  distance   li, 

from   the   surface   to  the   orifice  =  w7/,   and   the   kinetic 

energy  stored  up  in  w  as   it  issues  through    the  orifice 

w 
=  —  t'2  (Anal.  Mechs.,  Art.  217).     If  we  suppose  there  is 

no  loss  of  energy  during  the  passage  through  the  orifice, 

*  This  is  known  as  Torricelli's  Tlieorem. 

t  Tlie  term  head  in  Hydromechanics  is  measured,  relatively  to  any  point,  by  the 
depth  of  that  point  below  the  surface  of  the  liquid.  Since  the  liquid  in  Fig.  34 
descends  through  a  height  h  to  the  orifice,  we  may  say  there  are  h  feet  of  head 
above  the  orifice. 


138  VELOCITY  OF   EFFLUX. 

we  may  equate  these  two  quantities  of  work,  and  shall 
have 

from  which  we  find 

v^  =  2gh',  (1) 

/.    V  =  -v/p";  (2) 

that  is,  the  velocity  of  efflux  is  the  same  as  that  of  a 
body  which  has  fallen,  freely  through  the  height  h. 

From   (3)   we  have  A  =  — ,   in   which   the  height  h, 

corresponding  to  the  velocity  v,  is  called  the  head  due  to  the 
velocity,  or  simply  the  head.  The  corresponding  velocity  is 
called  the  velocity  due  to  the  head. 

Cor.  1. — If  the  orifice  be  made  in  the  vertical  face  of 
the  vessel,  and  a  tube  be  inserted  so  as  to  direct  the  current 
obliquely,  horizontally,  or  vertically  upicard,  the  velocity  of 
efflux  will  be  the  same,  since  the  pressure  of  fluids  at  the 
same  depth  is  the  same  in  every  direction  (Art.  7),  and 
each  particle  of  liquid  having  the  same  velocity  will  follow 
the  same  path ;  a  parabola  whose  directrix,  whatever  be  the 
angle  of  elevation,  is  fixed,  and  lies  in  the  surface  of  the 
liquid  (Anal.  Mechs.,  Arts.  151  and  153).  If  the  liquid 
issue  obliquely,  its  equation  is  given  in  (3)  of  Art.  151, 
Anal.  Mechs.  If  it  issue  horizontally,  «  =  0,  and  this 
equation  becomes 

sr  =  —  y  =  4:hy. 

Cor.  2. — If  Aj  be  the  depth  of  a  second  orifice  below  the 
Burface,  and  v^  the  velocity,  we  have 


v^  ==  V2gh^;  (3) 

therefore,  from  (2)  and  (3),  we  have 

v  :  Vj    ::   Vh  :  Vh^; 


VELOCITY  OF    EFFLUX.  139 

that  is,  the  velocities  of  efflux  are  as  the  square  roots 
of  the  depths. 

Cor.  3. — The  quantity  of  liquid  run  out  in  any  time  is 
equal  to  a  cylinder,  or  prism,  whose  base  is  the  area  of  the 
orifice,  and  whose  altitude  is  the  space  described  in  that 
time  by  the  velocity  acquired  in  falling  through  the  height 
of  the  liquid. 

Cor.  4. — If  any  pressure  be  exerted  on  the  surface  of  the 
liquid,  the  velocity  of  efflux  will  be  increased. 

Let  h  be  the  depth  of  the  orifice  below  the  surface  of  the 
liquid,  li^  the  height  of  the  column  of  liquid  which  would 
exert  the  same  pressure  as  that  which  is  applied  at  the 
surface;  then  the  velocity  of  efflux  will  be  due  to  the 
vertical  height  Ji  +  Ti-^;  hence  we  have  from  (2) 


v=y/^g{h  +  li^).  (4) 

If  h^  be  taken  equal  to  the  height  of  a  column  of  water 
equal  to  the  pressure  of  the  atmosphere  (^  34  feet),  (4) 

becomes  

V  =  ^/%g  {h  +  34).  (5) 

ivhich  is  the  velocity  of  efflux  ivhen  a  liquid  is  pro- 
jected into  a  vacuum,  the  oi^fice  bein^  at  a  depth  A, 
below  the  surface  of  the  liquid. 

If  k  be  the  area  of  the  orifice,  then  the  quantity  of  liquid 
Q  which  flows  through  the  orifice  in  the  unit  of  time  is 

Q  =  hv  =  k  V2gh.  (6) 

Cor.  5. — If  a  parabola,  with  a  parameter  =  2g,  be 
described  with  its  axis  vertical,  and  vertex  in  the  upper 
surface  of  the  liquid,  the  velocity  of  efflux  through  any 
small  orifices  in  the  side,  would  be  represented  by  the  cor- 
r<isponding  ordinates. 


140 


THE    HORIZONTAL    RANGE. 


ScH. — The  correctness  of  this  theorem  can  also  be  shown 
by  the  following  experiment.  If  in  the  vessel  (Fig.  34)  an 
orifice  K  or  R  be  made,  directed  vertically  upwards,  the 
velocity  of  the  jet  K  or  R  is  such  as  to  carry  the  particles 
of  liquid  up  ^nearly  to  the  same  level  as  the  surface  of  the 
liquid  in  the  vessel.  Practically  the  resistance  of  the  air 
and  friction  in  the  conducting  tube  destroy  u  portion  of 
this  velocity. 

EXAMPLES. 

1.  With  what  velocity  Avill  water  issue  from  a  small 
orifice  16^j  ft.  below  the  surface  of  the  liquid  ? 

Ans.  32|ft. 

2.  A  vessel  has  in  it  a  hole  an  inch  square;  water  is  kept 
in  the  basin  at  a  constant  level  of  9  ft.  above  the  hole; 
what  is  the  outflow  in  one  hour?  Ans.  QOO  cu.  ft. 

3.  What  is  the  discharge  per  second  through  an  orifice 
of  10  square  inches,  5  ft.  below  the  surface  of  the  liquid  ? 

Ans.  2152  cu.  ins. 


77.  The  Horizontal  Range  of  a  Liquid  Issuing 
through  a  very  Small  Orifice  in  the  Vertical  Side 
of  a  Tessel. — Let  ABOD  be  a  vessel  filled  with  a  liquid, 
having  its  side  BC  vertical,  M  a  small 

orifice  in  the  side  of  the  vessel,  MH     ^ jL 

the  parabola  described  by  the  liquid, 
and  CH  the  horizontal  range.  On 
BC  describe  the  semicircle  BFC,  and 
through  M  draw  MN  perpendicular 
to  BC.  If  the  liquid  issue  horizon- 
tally from  the  orifice  M,  the  equation 
of  its  path  is  (Art.  76,  Cor.  1), 


Fig.  35 


x^  =  4%,  (1) 

in  which  h  =  BM,  the  height  of  the  surface  above  the 


TIME    OF    DTSCHARGE.  141 

orifice ;  then  the  range  CH  will  be  determined  by  making 
y  =  MC.     Hence  we  have  from  (1) 


X  =  2  ^/hy  =  2  a/BM  X  MC 

=  2MN;  (2) 

that  is,  tJie  horizontal  range  of  a  liquid  issuing  horv- 
zontally  through  a  very  small  orifice  in  the  side  of  a 
vessel  is  equal  to  twice  the  ordinate  at  the  orifice,  in 
a  semicircle  whose  diameter  is  the  vertical  distance 
from  the  surface  of  the  liquid  to  the  horizontal 
plane. 

Cor. — When  the  orifice  is  made  at  the  centre  of  the  side 
BC,  the  horizontal  range  is  a  maximum,  and  equal  to  the 
height  of  the  liquid  above  CH;  at  equal  distances  above 
and  below  the  centre,  the  range  will  be  the  same. 

78.  Time  of  Discharge  from  a  Cylindrical  Vessel 

when  the  Height  is  Constant When  a  cylindrical 

vessel  is  kept  constanthj  full,  it  is  required  to  deter- 
mine the  time  in  ivhich  a  quantity  of  liquid  equal  in 
volume  to  the  cylinder  will  flow  through  a  small 
orifice  in  its  base. 

Let  h  be  the  height  of  the  surface,  K  the  area  of  the  base 
of  the  vessel,  and  k  of  the  orifice,  V  the  velocity  of  descent 
of  the  surface  of  the  liquid,  and  v  the  velocity  of  efflux  at 
the  orifice,  and  t  the  time  necessary  to  discharge  a  volume 
of  liquid  equal  to  that  of  the  cylinder,  which  remains  con- 
stantly full. 

Then  the  quantity  of  liquid  wiiich  flows  through  the 
orifice  in  the  unit  of  time  is  h  V^yh  ;  and  since  the  velocity 
of  the  surface  is  V,  the  quantity  of  liquid  which  passes 
through  the  orifice  in  the  unit  of  time  must  equal  VK. 
Hence  we  have 


142 


TIME   OF  EMPTYING   ANY   VESSEL. 


and  as  the  vessel  is  kept  constantly  full,  we  have 


,_  h__      hK 


Q 


{a) 


(1) 


where  Q  denotes  the  whole  quantity  of  liquid  in  the  vessel. 

•  Cor. — If  the  liquid  be  kept  at  a  height  //  in  a  second 
vessel,  containing  a  quantity  Q',  which  flows  through  an 
orifice  h',  in  the  time  t,  we  have  from  (1) 


/  =  --£ 


(2) 


kW^h' 
and  from  (1)  and  (2)  we  have 

Q  :Q'  ::   k  Vh  :  k'  VW. 

Hence,  the  quantities  discharged  in  the  same  time, 
from  orifices  of  different  sizes,  and  at  different 
depths,  are  as  the  areas  of  those  orifices  and  the 
square  roots  of  their  depths  jointly. 

79.  The  Time  of  Emptying  any  Vessel  through 
a  Small  Orifice  in  the  Bottom.— Let  EH  be  the  upper 

surface  of  the  liquid  at  the  time  /,  x  and 
y  the  distances  OD  and  DH,  Ti  the  depth 
OC  of  the  liquid  when  the  vessel  is  full, 
k  the  area  of  the  orifice,  and  K  the  area 
of  the  upper  surface  of  the  liquid  at  the 
time  t,  which,  when  the  figure  of  the 
vessel  is  known,  will  be  given  in  terms 
of  a;  and  y. 

Then  the  quantity  of  liquid  which  flows  through  the 
orifice  in  an  element  of  time  is  k  V^gx  dt ;  and  since  in  the 
same  time  the  surface  EH  descends  a  distance  dx,  the  quan- 


TIME   OF  EMPTYING   ANY  VESSEL.  143 

tity  of  liquid  which  flows  thrcdgli  the  orifice  in  this  time 
must  equal  Kdx.     Hence  we  have 

k  "s/^x  dt  =  —  Kdx, 

the  negative   sign   being  taken,  because  x  decreases  as  t 
increases, 

...     t  =  -f^^.  (1) 

^    k  V2gx 

Cor.  1. — If  the  vessel  be  a  surface  of  revolution  round  a 
vertical  axis.  A"  =  ny^,  which  in  (1)  gives 

t  =  -.r-t%.  (2) 

^  k  V^gx 

Cor,  2. — To  determine  the  time   of  emptying  a  right 
cylinder  or  prism.     Here  K  \&  constant,  and  (1)  becomes 


r  dx  2K      X    ,    ^ 

/   — -  =  — =  x^  +  O 


k  V2g       Vx  k  '^"Ig 

2A'    ,     „ 

remembering  tliat  wlien  ^  =  0,  x  z=z  h. 

When  a;  =  0,  we  have  for  the  time  of  emptying  the 
whole  cylinder, 

t  =  -^=Vh  =  -^=.>  (4) 

k  V^g  k  V2gh 

where  Q  denotes  the  quantity  of  liquid  in  the  vessel. 

By  comparing  this  result  with  that  in  (1)  of  Art.  78,  it 
appears  that  the  time  necessary  for  the  entire  discharge 
of  the  liquid  luhen  the  vessel  empties  itself  is  twice  as 
great  as  that  luhich  is  required  to  discharge  the  same 
quantity  when  the  vessel  is  kept  constantly  full^ 

Cor.  3. — If  a  cylinder  of  given  altitude  empty  itself  in 


144  THE  TIME   OF  EMPTYING  A    CYLINDER. 

n  seconds,  through  a  given  orifice,  the  radius  r  of  the 
cylinder  from  (4)  is 

.  /nk  Vg  .^. 

and  if  the  radius  is  given,  its  height  h  is 

^  -  2^'  ■  (^) 

80.  The  Time  of  Emptying  a  Cylinder  into  a 
Vacuum. — To  determine  the  time  in  which  a  cylin- 
drical vessel  ivill  empty  itself,  through  an  orifice  in 
the  bottom,  into  a  vacuum,  when  its  upper  surface  is 
exposed  to  the  pressure  of  the  atmosphere. 

Let  h  be  the  height  of  the  vessel,  h'  the  height  of  a 
column  of  liquid  which  is  equal  to  the  weight  of  the 
atmosphere  ;  and  x  the  depth  of  the  orifice  below  the 
upper  surface  of  the  liquid.  Then  from  (4)  of  Art.  76,  the 
velocity  of  discharge  is  's/^g  {x  +  It'),  which  in  (1)  of 
Art.  79  gives 

K       r       dx 


t  = 


k  '\/"2g 


2K 


[{h  +  A')^  -  (^  +  h'y^l    (1) 


k  V'ig 
since  when  x  =  h,  f  =:  0. 

And  making  x  =  0,  (1)  becomes 

t  =  ^^[ih  +  h')i-h'il  (2) 

which  is  the  time  of  emptying  the  vessel. 


CYLINDRICAL    VESSEL    WITH  TWO  SMALL   ORIFICES.   145 

81.  The  Time  of  Emptyiug  a  Paraboloid — Let 

the  vessel  he  a  paraboloid  of  revolution  round  the 
vertical  axis,  h  its  height,  and  2p  its  parameter.  Then 
if  X  is  the  depth  of  the  orifice  in  the  bottom  below  the 
upper  surface  of  the  liquid,  we  have 

y^  =  2pxy 

which,  in  (2)  of  Art.  79,  gives 

t  =  -  JP1=  /'?^  =  -  -J^^a^t  +  C 

since  when  x  =:  h,  t  =z  0. 

Making  a;  =  0,  and  putting  r  =  the  radiuas  of  the  base, 
(1)  becomes 

which  is  the  time  of  emptying  the  vessel. 

82.  Cylindrical  Tessel  with  Two  Small  Orifices. — 

A  cylindrical  vessel  of  given  dimensions,  is  filled  with 
a  liquid ;  there  are  two  given  and  equal  small  orifices, 
one  at  the  bottom,  the  other  bisecting  the  altitude ; 
to  find  the  time  of  emptying  the  upper  half,  suppos- 
ing both  orifices  to  be  opened  at  the  same  instant. 

Let  2a  =  the  altitude  of  the  vessel,  x  =  the  altitude  ol 
the  surface  of  the  liquid  from  the  upper  orifice  at  the  time 
t,  and  r  =  the  radius  of  the  base.  Then  the  quantities  of 
liquid  which  flow  through  the  upper  and  lower  orifices  in 
one  second  are,  respectively,  k  V^gx  and  k  V^g  {x  +  a), 
which  in  (1)  of  Art.  79,  gives 


146      ORIFICE  IN  THE  SIDE   OF  A    CONICAL    VESSEL. 


•nr' 


nr' 


27rr2 


/ 


dx 


k  's/'^g        ^/x  +  's/x  +  a 
^       ^  Va  +  X  —  ^/x 


dx 


[(2  Va  _  1)  at  _  (a  +  x)^  +  a;t],     (1) 


Zka  \/2^ 

oetween  the  limits,  x  =  a  and  x  =:  x. 
And  making  x  =  0,  (1)  becomes 


=^;?\/|(^^-^)' 


(3) 


which  is  the  time  of  emptying  the  upper  half  of  the 
vessel.     (See  Bland's  Hydrostatics,  p.  165.) 

83.  Orifice  in  the  Side  of  a  Conical  Tessel.— .i 

hollow  cone,  base  downward,  whose  vertical  angle  is 
60°,  is  filled  ivith  a  liquid;  to  determine  the  place 
where  a  small  orifice  must  he  made  in  its  side,  so  that 
the  issuing  liquid  may  strike  the  horizontal  plane  in 
a  point  ivhose  distance  from  the  bottom  of  the  vessel 
is  to  the  distance  of  the  orifice  from  the  top  : :  5  :  4- 

Let    AN  =  X,    and    AM 

then  NM  =  a — x,  and  AP  = 

Also  by  hypothesis  we  have 

BO  :  AP  : :  5  :  4 ; 

...    B0=-^ 
2Vd 


BR  =  PR  tan  30°  = 


a  —  a? 


a/3  2  V3 


V3    ' 

2a  +  Sx 

3  a/3 


VELOCITY  OF  EFFLUX. 


147 


Substituting  tliis  value  of  RO  for  x  in   (3)  of  Art.  151, 
Anal.  Mechs.,  and  for  i^  its  value  2^a:,  we  have 


o  _  /2a  +  Sa^Y 


4.x  cos2  30° 


2a  +  Sx  ,       ^^ 

X  —  a  =  -—  tan  30 

2\/3 

_  2a  +  Sx  _  (2a  +  3x)\ 

~        6  36x        ' 

.:    x=  la{l  ±Vl); 

ivhich  is  the  depth  of  the  orifice  from  the  vextex.    (See 
Bland's  Hydrostatics,  p.  142.) 

84.  Telocity  of  Efflux  through  au  Orifice  of  any 
Size  in  the  Bottom  of  a  Cylindrical  Vessel.— Let 

AB  be  the  upper  surface  of  the  liquid  at  a 
height  h  above  the  orifice  EF;  consider  any 
lamina  GH,  at  a  height  x  above  the  orifice; 
and  as  before,  let  k,  K  be  tlie  areas  of  the 
orifice  and  the  section  GH.  respectively. 

At  the  height  x  above  the  orifice,  let  p  be 
the  pressure  and  p  the  density,  and  at  a  height 
X  4-  dx,  let  p  -\-  dp  he  the  pressure.  Then 
the  volume  Kdx  of  liquid  may  be  considered 
as  acted  upon  by  the  pressures  pK,  {p  +  dp)  Ky  and  its 
weight,  —  gpKdx.     Hence  the  moving  force  will  be 

pK  —  {p  ■\-  dp)  K  —  (jpKdx  =  —  Kdp  —  gpKdx  ;* 

and  since  the  moving  force  is  measured  by  the  mass  into 
the  acceleration  (Anal.  Mechs.,  (3)  of  Art.  20),  we  have 

d^x 


—  Kdp  —  fjpKdx  =  pKdx  -Tj^ , 


(Px 
df 


dp  +  gpdx 
pdx 


(1) 


*  The  moving  force  is  here  negative  because  x  is  positive  upwards. 


148  VELOCITY  OF  EFFLUX. 

Let  V  be  the  velocity  of  efflux  ;  then  the  quantity  ol 
liquid  which  flows  through  tlie  orifice  in  an  element  of 
time  is  kvdt,  and  since  in  the  same  time  the  surface  K 
descends  a  distance  equal  to  dx,  we  have 

Kdx  =  —  kvdf,     or    ^  = t?,  (2) 

dt  Ji. 

the  negative   sign   being  taken   because  x  decreases  as  t 
increases ; 

d^ k  dv 

''•     d^  ~  ~Kdi' 
which  in  (1)  gives 

k  dv  dp  +  gpdx 

K  dt  pdx 

or, 

,  ,  pkdv  dx  pk^        ,      rn  /«\T 

dp  +  gpdx  =  -^  —=  -—vdv  [from  (2)]. 
Integrating,  we  have 

pv^  /,        P  \ 

remembering  that  when  x  =i  0,  Ji  =  k. 
Hence,  2gx  =  v^(l  —  -j-j; 


...  .  =  ^-^i 

*^    1  — 


2gx 

^2 


(3. 


which  is  the  velocity  of  efftux  at  a  depth  x. 

When  X  =  li,  or  the  vessel  is  full,  we  have  for  the  velocity 


EXAMPLE.  149 

Cob.  1. — As  the  ratio  -^  of  the  sections  decreases  the 
A 

velocity  decreases,  becoming  a  minimum  and  =  "^"Zgli, 
when  the  cross-section  k  of  the  orifice  is  very  small  com- 
pared with  that  of  K,  which  agrees  with  (2)  of  Art.  76,  as 
it  clearly  should. 

Tc 
Cor.  2. — As  the  ratio  -77  increases  the  velocity  increases, 

and  it  approaches  nearer  and  nearer  to  infinity,  the  smaller 
the  difference  between  the  two  cross-sections  becomes.  If 
h  =^  K,  (4)  becomes 

from  which  we  infer  that,  if  a  cylindrical  vessel  is  without 
a  bottom,  a  liquid  must  flow  in  and  out  with  an  infinitely 
great  velocity,  or  else  a  section  of  the  liquid  flowing  out  of 
the  vessel  can  never  be  equal  to  a  section  of  the  vessel.  If 
a  cylindrical  tube  be  vertical,  and  filled  with  a  liquid,  the 
portion  of  the  liquid  at  the  lower  extremity,  being  urged  by 
the  pressure  of  all  above  it,  will  necessarily  have  a  greater 
velocity  than  those  portions  which  are  higher,  and  therefore 
(Art.  ?5)  a  section  of  the  liquid  issuing  from  the  vessel 
must  be  less  than  a  section  of  the  tube,  *.  e.,  the  stream  of 
liquid  will  not  fill  the  orifice  of  exit.* 

EXAMPLE. 

If  water  flows  from  a  vessel,  whose  cross-section  is  60 
square  inches,  through  a  circular  orifice  in  the  bottom 
5  inches  in  diameter,  under  a  head  of  water  of  6  feet,  find 
its  velocity.  Ans.  20.79  ft. 

85.  Rectangular  Orifice  in  the  Side  of  a  Vessel.— 

To    determine    the    quantity    of   liquid    which   will 

*  Formula  (4)  was  first  given  by  Beruoullli,  and  was  afterwards  much  disputed 
(Weisbach^s  Mechanics,  p.  804). 


150 


RECTANGULAR    ORIFICE  IX  A    VESSEL. 


flow  fvuin  a  rectangular  orifice  in  the  side  of  a  vessel 
which  is  hept  constantly  full. 

(1)  Wlien  one  side  of  the  orifice  coincides  with  the 
surface  of  the  liquid. 

Let  h  be  the  height  and  b  the 
breadth  of  the  rectangular  ori- 
fice ALMD,  through  which  the 
efflux  takes  place  ;  let  HK  be 
a  horizontal  strip  at  the  dis 
tance  x  below  AD,  and  of  infin- 
itesimal thickness  d.r,  so  that 
the  velocity  of  the  liquid  in 
every  part  of  the  strip  is  the 
same. 

Then  the  velocity  of  efflux  through  this  strip  is  V^5'^ 
[Art.  76,  (2)],  and  the  quantity  discharged  in  a  unit  of 
time  is  hdx  's/^gx',  hence,  caUing  Q  the  whole  quantity 
discharged  in  a  unit  of  time,  we  have 


Q  =^    I  hdx  V^gx ; 


(1) 


and  integrating  between  a;  =  0  and  x  =  h,  we  have 

Q  =  ^b  V2^^  (2) 

If  we  denote  by  v  the  mean  velocity,  i.  e.,  the  velocity 
which  would  have  to  exist  at  every  point  of  the  orifice,  i7i 
order  that  the  same  quantity  of  liquid  would  fldw  thronuli 
the  orifice  with  a  uniform  velocity  as  now  flows  through 
with  the  variable  velocity,  we  have 


which  in  (2)  gives 


V  = 


(3) 


Hence   thC'  mean   velocity   of    a    liquid    flowing  out 
through  a  rectangular  orifice  in  the  side  of  a  vessel 


TRIANGULAR  ORIFICE  IN  THE  SIDE  OF  A   VESSEL.    151 

is  I  the  velocity  at  the  lower  edge  of  the  orifice ;  and 
the  quantity  of  liquid  flowing  out  through  this  orifice 
in  any  given  time  is  f  the  quantity  that  tvould  flow 
through  an  orifice  of  equal  area  placed  horizontally 
at  the  whole  depth,  in  the  same  time,  the  vessel  being 
kept  constantly  full. 

(2)  WJien  tlie  upper  surface  of  the  rectangular  ori- 
fice is  beloiv  the  surface  of  the  liquid. 

Let  SR  be  the  upper  edge  of  the  orifice  at  the  depth' /ii 
below  the  surface  AD.  Then,  integrating  (1).  between  the 
limits  X  =z  h^  and  x  =  h,  we  have 

Q  =  ^bV2^{hi-hfy  (4) 

If  the  mean  velocity  of  efflux  is  v,  we  have 

Q  =  b{h-h,)v, 


which  in  (4)  gives 


-^^^feir*-  <^> 


86.  Triangular  Orifice  in  the  Side  of  a  Tessel.— 

(1)    When  the  vertex  of  the  triangle  is  in  the  surface 
of  the  liquid. 

A_K EJ) 

Let  h  be  the  height  EF,  and  h  the  breadth 

HF  of  the  triangular  orifice  EHF,  through 
which  the  efflux  takes  place;  let  LM  be  a 
horizontal  strip  at  the  distance  x  below  AD, 
and  of  infinitesimal  thickness  dx,  so  that  the 
velocity  of  the  liquid  in  every  part  of  the  strip 
is  the  same. 

Then  LM  =  t  a:,  and  calling  Q  the  quantity  of  liquid 

discharged  in  a  unit  of  time,  we  have 


152    TRIANG  ULAR  ORIFICE  IN  THE  SIDE  OF  A    VESSEL. 


Q  =i  I    J  x'^lgx  dx 

If  the  mean  velocity  is  v,  we  have 
Q  =  \bhVy 
which  in  (1)  gives  v  =  ^^/'ii,gh. 


(1) 


(2) 


(2)  When  the  base  of  the  triangle  is  in  the  surface 
of  the  liquid. 

Let  KEH  be  the  triangular  orifice,  KE  =  5,  and  KH 
=  h.  Then  the  quantity  discharged  through  KEH  will 
equal  the  discharge  through  the  rectangle  KHFE,  minus 
that  through  the  triangle  EHF ;  therefore  subtracting  (1) 
of  this  Art.  from  (2)  of  Art.  85,  we  have 


and 


(3) 
(4) 
AFKE    B 


CoE.  1. — If  the  orifice  be  a  trapezoid  ABCD, 
whose  upper  base  AB  =  b^  lies  in  the  sur- 
face of  the  liquid,  whose  lower  base  CD  =  b^, 
and  whose  altitude  is  DF  =  h,  the  discharge 
may  be  found  by  combining  the  discharge 
through    the    rectangle    ECDF    with     those  Fig-  4i 

through   the  two  triangles   ADF  and  BCE. 
Hence,  combining  (3)  with  (2)  of  Art.  85,  we  have 


Q  =  ^b^hV^h-\-h  (*i  -  *2)  ^'V25rA 

=  A(2*i  +^h^)hV^h.  (5) 


TKlAyGtlLAR  ORIFICE  IN  THE  SIDE  OF  A   VESSEL.   153 


Cob.  2. — If  the  orifice  be  a  triangle  DCH  (Fig.  41), 
whose  base  DC  =  b^  is  situated  at  a  depth  KL  =  /(j  be- 
low the  surface  of  the  liquid,  and  whose  vertex  H  is  at  a 
depth  h  below  the  surface,  the  discharge  is  equal  to  that 
through  the  triangle  AHB,  minus  that  through  the  trap- 
ezoid ABCD.     Hence,  from  (3)  and  (5),  we  have 


Q  =  ^^bhV2ffh  -  ^K{2b  +  Sb,)  h,V2gh^ 

=  ^W2g{2b{hi  -  h,^)  -  3b,h,il 

Since  AB  :  DC  ::  HK  :  HL,  we  have 
b  '.  by  ::  h  '.  h  —  A^  ; 


(6) 


which  in  (6)  gives 


Q  = 


_  2\/2gb^  (-ilfi  —  S/J/^t  +  37fJ 


15 


h  -  h^ 


(7) 


Cor.  3. — If  the  orifice  be  a  triangle  ABC,       I 
whose  vertex  A  is  above  its  base,  and  at  a     . 
depth  /ii  below  the  surface  of  the  liquid, 
whose  base  CB  =  Ji  is  at  a  depth  h  below 
the  surface,  the  discharge  is  equal  to  that 
through  the  rectangle  ACBK,  minus  that     H 
through  the  triangle  ABK.    Hence,  from 
(7)  and  (4)  of  Art.  85,  we  have 


Fig.  42 


Q  = 


2V¥g  b^  /Sh^  —  5h^hi  +  2h^h        '  ,gx 


15 


154 


ORIFICE  IN  THE  SIDE   OF  A    VESSEL. 


Otherwise  thus :  Let  ODC  be  a  vertical  orifice,  formed  by 
a  plane  curve,  whose  vertex  is  0,  at  the 
depth  AO  below  the  surface  of  the  liquid. 

Let  AB  =  h,  AO  =  h^,  OE  =  a;,  EQ  = 
y  ;  then  the  area  of  the  horizontal  strip  PQ, 
of  infinitesimal  thickness  dx,  =z2y  dz  ;  and 
therefore  the  quantity  dischai'ged  in  a  unit 
of  time  through  this  elemental  strip  is 


H 

f 

0 

2y  dx  V2g  {h^  +  x) ; 
and  hence  we  have 


B 

Fig.  43 


Q  =  J  2yV2g  (7^1  +  x)  dx. 


(9) 


(1)   When  the  orifice  is  a  rectangle. 

Here  y  is  constant,  which  put  =  ^h,  and  integrating  (9) 
between  the  limits  x  =  0  and  a;  =  //  —  h^,  we  have  for 
the  discharge  through  the  whole  orifice  ODC, 


which  is  the  same  as  (4)  of  Art.  85. 


(10) 


Cor.  4. — If  the  upper  side  coincides  with  the  surface  of 
the  liquid,  h^  ==  0,  and  (10)  becomes 

Q  =  ^bhV2^h, 

which  agrees  with  (2)  of  Art.  85. 

(2)    When  the  orifice  is  a  triangle  whose  vertex  is 
downwards  and  the  hase  horizontal. 

Let  a  :  5  be  the  ratio  of  the  altitude  to  the  base;  then 

2y  =  -  (^i  -  ^h  —  «)» 

which  in  (9),  and  integrating  between  the  limits  a;  =  0  and 
X  =^  h  —  h^,  gives 


ORIFICE  IN  THE  SIDE   OF  A    VESSEL.  155 

Q  =^  J  -a/2^  {h  —  7^1  —  x)  VA J  +  xdx 

which  agrees  with  (7). 

CoK.  5. — If  the  base  coincides  with  the  surface  of  the 
liquid,  h^  =■  0,  and  (11)  becomes 

which  agrees  with  (3). 

(3)  WTieri  the  orifice  is  a  triangle  whose  vertex  is 
upwards  and  base  horizontal. 

Here  2v  =  -  a:, 

which   in    (9),   between   the   same  limits,  a;  =  0  and  x  = 
Ti  —  h^,  gives 

Q  =  ^W^,i^A^f^±lb^,  (1,) 

which  agrees  with  (8), 

Cor.  6. — If  the  vertex  coincides  with  the  surface  of  the 
liquid,  Aj  =0,  and  (12)  becomes 

Q  =  \hhV^h, 
which  agrees  with  (1). 

Cor.  7. — From  Cors.  5  and  6  we  see  that  the  quantities 
discharged  in  the  same  time  through  two  equal  triangular 
orifices  in  the  side  of  a  vessel  kept  constantly  full,  the  one 
having  its  base  and  the  other  its  vertex  upwards  in  the  sur- 
face of  the  liquid,  are  in  th?  ratio  of  2  :  3. 


156 


EXAMPLE. 


87.   The  Time  of  Emptying  any  Tessel  through 
a  Vertical  Orifice. — Let  A  be  the  surface  ^ 

of  the  liquid  in  the  vessel  when  the  orifice 
OCD  is  opened,  and  H  the  surface  at  the  end 
of  the  time  t ;  let  AH  =  z,  k.0  =  h',  OE 
=  r,  AB  =  h,  and  PQ,  =  2y. 

Then  the  quantity  discharged  through  the 
orifice  in  an  element  of  time,  from  (9)  of 
Art.  86,  is 


H 

p/" 

0 

/ 

^  \ 

Q  = 


2a/3W  yVx  +  h'—z  dx 


D       B       C 

Fig.  44 


dt, 


(1) 


the  ar-integration  being  taken  between  h  —  h'  and  0,  z 
being  constant  during  this  integration;  and  since,  in  the 
same  time,  the  surface  of  the  liquid  at  H  descends  a  dis- 
tance dz,  the  quantity  discharged  through  the  orifice  in 
this  time  must  equal  Kdz,  where  K  is  the  area  of  the  sec- 
tion of  the  vessel  at  H.     Hence,  we  have 


2V2gJ  yVx  +  h' 

■  '  =  4=/- 


z  dx 
Kdz 


dt  =  Kdz\ 


2^/'Zg*J  fy^/x  -\-  h'  —  z  dx 
the  ^-integration  being  taken  between  0  and  h. 


(2) 
(3) 


EXAMPLE. 

Find  the  time  of  emptying  a  cone 
by  an  orifice  ACB  in  its  side. 

Let  AH  =  A  be  the  axis  of  the  cone, 
CB  =  h,  CA  =  I,  angle  HAC  =  a, 
AK  =  X,  PK  being  perpendicular  to 
AH.  When  the  orifice  is  opened,  let 
the  surface  of  the  liquid  in  the  vessel 
be  at  H,  and  at  the  end  of  the  time  t 
let  it  be  at  M,  and  let  AM  =  z. 


EXAMPLE. 


167 


Then  we  have 

AP  =  X  sec  «, 


Pp  =  sec  a  dx, 


nn>         ^  sec  a 
and  y  =  ^?   =  — ^ —  x; 

/.     the  area  of  PP;j  j9  = ^ —  .^•  aa;. 

The  velocity  of  discharge  through  this  area 
=  y/^g  {z  -^) ; 
therefore  the  quantity  discharged  in  an  element  of  time 

b  sec^  a     —  /*      ' \ 


I 
b  sec^ 


'\/2(j  I  xy/z  —  xdx 


dt 


I 

the  a;-limits  being  0   and  z;    and  this  must  equal  Kd&, 
from  (2). 

Hence  we  have,  from  (3),  taking  the  negative  sign,  be- 
cause z  decreases  as  t  increases, 


=  / 


—  Kdz 


b  sec^a 

T~ 


y/^^z^ 


—  15?7r  tan'-* «  ^  dz 
^by/^g  sec^  a  ^ 

15?7r  tan^tt     Pdz 
4&V^sec8«'^  z^ 

15Tr?  tan2 « 


Uy/^g  sec2 « 
between  the  limits  h  and  z. 


W'h  -  V4 


158  EFFLUX  FROM  A    VESSEL   IN  MOTION. 

Therefore  the  whole  time  of  emptying  the  vessel 

15t?  tan^  «  \/7i 

'2^hy/%g  sec^  a 

(See  Bland's  Hydrostatics,  p.  185.) 

88.  Efflux  from  a  Yessel  in  Motion. — If  the  vessel 
ABCD  be  filled  with  liquid  to  AB,  and  raised  vertically, 
with  an  accelerated  motion,  by  a 
weight  P  attached  to  an  inextensible 
string,  without  weight,  passing  over 
two  smooth  pulleys  F,  E,  the  veloc- 
ity of  efflux  is  augmented ;  and  if  it 
descends  with  an  accelerated  motion, 
the  velocity  is  diminished. 

Let  Q  be  the  weight  of  the  vessel 
and  liquid  contained  in  it.  Since 
the  pulleys  are  perfectly  smooth,  the 
tension  of  the  string  is  the  same 
ti)roughout ;  hence  the  force  which  causes  the  motion  is 
the  difference  between  the  weights  P  and  Q.  The  moving 
force,  therefore,  is  P  —  Q\  but  the  weight  of  the  mass 
moved  is  P  -\-  Q.  Hence,  from  (1)  of  Art.  25,  Anal. 
Mechs.,  we  have 


9 

J  -  P  ^  Qy> 


(1) 


which  is  the  vertical  force  of  acceleration.  Since  this  force 
acts  vertically  upwards  on  the  vessel,  and  the  force  of  grav- 
ity ^  acts  vertically  downwards,  every  particle  of  the  liquid 
presses  against  the  bottom  of  the  vessel,  not  only  with  its 
own  weight  Mg,  but  also  with  its  inertia  M/.    Hence  the 


EFFLUX  FROM  A    VESSEL  IN  MOTION.  159 

entire  accelerating  force  pressing  against  every  point  in  the 
base  is 

9+f=9-^T~^Q9  [from(l)] 

^9-  (2) 


Let  HO  =  h,  and  v  =  the  velocity  of  efflux  ;  then  we 
have 

V  =  VHf+7)h  (3) 


Cor.  1. — If  the  vessel  is  allowed  to  empty  itself  through 
the  orifice  0,  without  receiving  any  liquid,  let  x  =  the 
variable  altitude  OH,  K  the  horizontal  section  of  the  vessel, 
which  is  a  function  of  x,  and  k  the  section  of  the  orifice. 
Then  we  have 

Q  =  fxdx ; 

which  in  (4)  gives  for  the  quantity  discharged  in  an  element 
of  time. 


k\/'Zg\/  -p-T-TT^FTT;.  dt  =  —  Kdx\ 


fKdx 
r—  KdxVP  +  fKdx  ,^. 

.*.        t    =       /    ;= , (5) 

"^  kV2g'V2Px 

Cor.  2. — If  f  =  g,  (3)  becomes 

V  =  V2-2gh  =  2Vgh; 

and  the  velocity  of  efflux  is  1.414  times  as  great  as  it  would 
be  if  the  vessel  stood  still. 


160 


EFFLUX  FROM  A   ROTATING    VESSEL. 


Cor.  3.— If  in  (1),  P  =  Q,  then  /  =  0,  and  the  vessel 
is  at  rest.  If  P  <C  Q,  then  Q  will  descend  and  P  ascend, 
/  is  negative  and  (3)  becomes 

and  the  vessel  descends  with  an   accelerated   motion,  the 
velocity  being  diminished. 

Cor.  4.— If  P  =  0,  then,  from  (2),  g+f=  0,  and 
therefore,  from  (3),  v  =z  0,  and  there  is  no  pressure  on  the 
bottom  of  the  vessel,  and  no  liquid  will  flow  out;  which  is 
also  evident  from  this,  that  every  particle  in  the  vessel  will 
descend  by  its  own  gravity,  Avith  the  same  velocity. 

89.  Efflux  from  a  Rotating  Yessel. — If  a  vessel 
ABCD,  containing  a  liquid,  is  made  to  rotate  about  its  ver- 
tical axis  XX',  the  surface  of  the  liquid 
will  take  the  form  of  a  paraboloid  of  revo- 
lution (Art.  21),  and  at  the  centre  H  of  the 
bottom  the  depth  of  liquid  KH  is  less  than 
it  is  near  the  edge,  and  the  liquid  will  flow 
more  slowly  through  an  orifice  at  the  centre 
than  through  any  other  orifice  of  the  same 
size  in  the  bottom. 

Let  h  denote  the  height  KH ;  then  the 
velocity  of  efflux  through  an  orifice  at  H  = 
'^'Igh.  Let  y  denote  the  distance  HO  = 
MP  of  an  orifice  0  from  the  axis  XX',  and  «  the  angular 
velocity ;  then,  since  the  subtangent  MT  is  bisected  at  K, 
we  have,  for  the  height  of  the  liquid  at  P  above  the  centre  K, 

KM  =  ^TM  = 
MP 


=  \y 


MN 


=  ^  [from  (2)  of  Art.  21]. 


THE    CLEPSYDRA,    OR    WATER- CLOCK.  161 

Hence,  the  velocity  of  efflux  through  the  orifice  at  0  is 


v  =  ^^g{^u  +  y^\ 


%g1 
=  ^/'Igh  +  fd^.  (1) 

ScH. — This  formula  is  true  for  a  vessel  of  any  shape,  even 
when  it  is  closed  at  the  top  so  that  the  paraboloid  AKB 
cannot  he  completely  formed.  In  this  case  also,  7i  is  the 
depth  of  the  orifice  below  the  vertex  K,  and  ^/w  is  the  veloc- 
ity of  rotation  of  the  orifice.  (See  "Weisbach's  Mechs., 
p.  819.) 

90.   The  Clepsydra^   or  Water-Clock.— This  is  ar. 

instrument  consisting  merely  of  a  vessel  from  which  the 
water  is  allowed  to  escape  through  an  orifice  in  the  bottom, 
and  the  intervals  of  time  are  measured  by  the  depressions  of 
the  upper  surface.  Thus,  if  we  wish  the  clock  to  run  12 
hours,  we  let  ^  =  13  hours  =  12  x  60  x  60  seconds;  then 
solving  (4)  of  Art.  79  for  h,  we  have 

and  substituting  in  it  this  value  of  t,  we  have 

A  =  1^(12x60x60)8, 

which  gives  the  depth  of  liquid  in  the  cylindrical  vessel  that 
will  empty  itself  in  12  hours. 

(1)  To  discover  the  manner  in  which  the  height  h  of  the 
vessel  must  be  divided  in  order  that  the  upper  surface  of 
the  liquid  may  descend  through  the  several  divisions  of  the 
scale  in  equal  intervals  of  time,  we  make  tva  (1)  successively 

equal  to  12,  11,  10, 4,  3,  2,  1  hours,  and  get  for  h  a 

series  of  values  which  are  as  144,  121,  100, .  .  .  16,  9,  4,  1 ; 
hence,  if  the  height  h  be  divided  into  144  equa/  spaces,  and 


162  THE    VENA    CONTRACTA. 

marked  upwards  from  the  bottom  of  the  vessel,  then  the 

marks  121,  100, 16,  9,  4,  1,  0,  will  give  the  water  level 

at  1,  2, 8,  9,  10,  11,  12  hours  after  the  water  begins  to 

flow. 

(2)  Any  vessel  may  serve  for  a  clepsydra,  but  that  form  is 
most  convenient  in  which  the  upper  surface  of  the  liquid 
descends  uniformly. 

Let  X  =  the  height  of  the  liquid  in  the  vessel,  K  the 
area  of  the  descending  surface,  v  its  velocity,  and  k  the  area 
of  the  orifice.     Then  from  (a)  of  Art.  78,  we  have 

V  =  -^V^gx.  (2) 

And  since  the  surface  is  to  descend  uniformly,  this  value 
of  V  must  be  equal  to  some  constant  a,  which  will  depend 
upon  the  whole  height  and  the  time  in  which  the  clepsydra 
will  be  emptied  ;  hence  (2)  becomes 

K^  =  ~^^-;  (3) 

and  supposing  the  area  of  the  descending  surface  of  the 
liquid  to  be  a  circle  =  rryS^  (3)  becomes 

which  is  a  parabola  of  the  fourth  order. 

Hence,  the  heights  of  the  sections  must  vary  as  the 
fourth  power  of  their  radii. 

91.  The  Vena  Con tr acta. — The  laws  of  efflux  that 
have  been  deduced  are  founded  on  the  hypothesis  that  the 
liquid  particles  descend  in  straight  lines  to  the  orifice,  and 
»11  issue  in  parallel  lines  with  a  velocity  due  to  the  height 


COEFFICIENT   OF  CONTRACTION.  163 

of  the  liquid  surface.  Experiment  shows,  however,  that 
this  is  not  the  case.  The  liquid  does  not  issue  in  the  form 
of  a  prism,  and  hence  the  quantity  discharged  in  a  unit  of 
time  is  not  measured  by  the  contents  of  a  prism  whose  base 
is  the  orifice  and  whose  altitude  is  the  velocity;  this  would 
give  the  theoretical  discharge  (Art.  76,  Cor.  3),  but  the  prac- 
tical discharge  is  generally  much  less.  When  a  vessel 
empties  itself  through  an  orifice,  it  is  observed  that  the 
particles  of  liquid  near  the  top  descend  in  vertical  lines ; 
but  when  they  approach  the  bottom  they  take  a  curvilinear 
course,  being  turned  in  towards  the  orifice,  or  spirally 
around  it,  and  this  deviation  from  a  vertical  rectilinear  path 
is  the  greater  the  further  the  horizontal  distance  of  the 
particles  is  from  the  orifice.  The  oblique  direction  of  the 
exterior  particles  within  the  vessel  continues  through  the 
orifice,  and  gives  the  stream  of  liquid,  in  issuing'  from  the 
orifice,  nearly  the  form  of  a  truncated  cone  or  })yramid, 
whose  larger  base  is  the  area  of  the  orifice.  This  diminu- 
tion in  the  size  of  the  issuing  stream  is  called  the  contrac- 
tion of  the  vein,  and  the  section  of  the  stream  at  the  point 
of  greatest  contraction  is  called  the  Vena  Contracta,*  or 
contracted  vein. 

From  the  results  of  most  experiments,  the  vena  contracta, 
when  the  orifice  is  a  circle,  is  at  a  distance  from  the  orifice 
equal  to  the  radius  of  the  orifice. 

92.  CoeiBcient  of  Contraction. — When  water  flows 
through  orifices  in  thin  plates,  it  has  been  found,  by  meas- 
urements of  the  stream,  made  by  different  experimenters, 
that  its  diameter  at  the  vena  contracta  is  about  0.8  of  the 
diameter  of  the  orifice.  The  ratio,  therefore,  of  the  cross- 
section  of  the  vena  contracia  to  that  of  the  orifice  in  a  thin 


♦  This  name  was  first  given  by  Newton,  who  also  showed  that,  by  taking  the 
area  of  the  vena  contracta  as  the  area  of  the  orifice,  and  rejrarding  the  height  of  the 
surface  above  the  vena  contracta  as  the  height  of  the  vessel,  the  theoretic  discharge 
agreed  far  more  closely  with  the  practical. 


164  COEFFICIENT  OF  EFFLUX. 

plate  is  0.64.  This  ratio  is  called  the  Coefficient  of  Contrac- 
tion.*  Denoting  it  by  «,  we  have  ak  for  the  section  of  the 
vena  contractu,  h  being  the  section  of  the  orifice  (Art.  76). 
Substituting  ah  for  k  in  (6)  of  Art.  76,  we  have,  for  the 
quantity  Q^  discharged, 

Q^  =  ahv  z=  t(kV2gk  =  MkV2gh,  (1) 

which  is  the  quantity  discharged  in  a  unit  of  time. 

93.  Coefficient  of  Velocity. — The  actual  velocity  of 
discharge  is  found  by  experiments  to  be  a  little  less  than  the 
theoretical  velocity,  v  =  '\/2gh.  Experiments  f  made  with 
polished  orifices  have  shown  that  the  actual  velocity  is  from 
96  to  99  per  cent,  of  the  theoretical  one.  This  loss  of  ve- 
locity arises  from  the  friction  of  the  water  upon  the  inner 
surface  of  the  orifice,  and  from  the  viscosity  of  the  water. 
The  ratio  of  the  actual  velocity  to  the  theoretical  velocity  is 
called  the  Coefficient  of  Velocity.  This  coefficient  is  found 
to  be  tolerably  constant  for  different  heads  with  well-formed 
simple  orifices,  and  it  very  often  has  the  value  0.97.  De- 
noting the  coefficient  of  velocity  by  (f>,  and  the  actual 
velocity  by  v^,  we  have 

v^  r=  <f>v  =  (pV2gh  =  .97'v/2p',  (1) 

which  is  the  actual  velocity  of  efflux. 

94.  Coefficient  of  Efflux.— If  the  yalue_of  v,  in  (1) 
of  Art.  93  be  substituted  for  the  velocity  V'^gh  in  (1)  of 
Art.  92,  we  have,  for  the  actual  discharge  Q^, 

Qg  =  tc]c(pv  =  ah(f>V2glt 

=  .64  X  .9'7kV2gfi  =  .62kV2gh.  (1) 

•  This  ratio  is  not  constant,  but  undergoes  variations  by  varying  the  form  of  the 
orifice,  the  thickness  of  the  surface  in  which  the  orifice  is  made,  or  the  form  of  th« 
vessel. 

t  Experiments  made  by  Michelotti,  Eytelwein,  and  others. 


EFFLUX  THROUGH  SHORT  TUBES,    OR  AJUTAGES.    165 

The  ratio  of  the  actual  discharge  Q^  to  the  theoretical 
discharge  Q  is  called  the  Coefficient  of  Efflux. 

Denoting  the  coefficient  of  efflux  by  /-i,  we  have,  from  (1) 
and  (6)  of  Art.  76, 

i^  =  ^  =  ««/>  =  .62;  (2) 

i.  e.,  the  coefficient  of  efflux  is  the  product  of  the  coef- 
ficient of  velocity  and  the  coefficient  of  contraction. 

ScH. — The  value  of  fi  can  also  be  determined  by  direct 
measurement  of  the  discharge  in  a  given  time,  an  observa- 
tion which  can  be  made  with  much  greater  accuracy  than 
those  of  contraction  and  velocity,  on  which  it  depends.  In 
the  present  case  it  is  found  by  direct  measurement  to  be  .62, 
agreeing  well  with  the  product  .64  x.97,  of  the  values  above 
given.* 

Rem. — Repeated  observations  and  experiments  have  led  to  the  con- 
clusion that  the  coefficient  of  efflux  is  not  constant  for  all  orifices  in 
thin  plates  ;  that  it  is  greater  for  small  orifices  and  small  velocities  of 
efflux  than  for  large  orifices  and  great  velocities,  and  that  it  is  much 
greater  for  long  narrow  orifices  than  for  those  whose  forms  are  regu- 
lar or  circular.  For  square  orifices,  whose  areas  are  from  1  to  9  square 
inches,  under  a  head  of  from  7  to  21  feet,  according  to  the  experiments 
of  Bossut  and  Michelotti,  the  mean  coefficient  of  efflux  m  fi  —  .610  ; 
for  circular  orifices  from  -|  to  6  inches  in  diameter,  with  from  4  to  20 
feet  head  of  water,  it  is  //  =  .615,  or  about  ^.f 

95.  Efflux  through  Short  Tubes,  or  Ajutages. — 

If  the  water,  instead  of  flowing  through  an  orifice  in  a  thin 
plate,  be  allowed  to  discharge  through  short  tubes,  called 
also  ajutages  and  mouth-pieces,  the  quantity  discharged  from 
a  given  orifice  is  considerably  increased.  More  seems  to  be 
gained  by  the  adhesion  of  the  liquid  particles  to  the  sides  of 
the  tube,  in  preventing  the  contraction  of  the  stream,  than 
is  lost  by  the  friction.     Ajutages  of  different  forms  have 

•  Cotterill'8  Applied  Mechs.,  p.  449. 

+  Weisbach's  Mechs.,  p.  824 ;  also,  Tate's  Mech.  Phil.,  p.  388. 


166    h'FFLUX  THROUGH  SHORT  TUBES,    OR   AJUTAGES. 

different  degrees  of  advantage  in  this  respect,  which  cm  be 
determined  only  by  experiment.  The  discharge  is  found  to 
be  greater  when  the  ajutage  is  conical  and  the  larger  end 
is  the  discharging  orifice. 

(1)  The  results  of  many  experiments*  made  with  cylin- 
drical tubes  1\  to  3  inches  in  diameter,  the  length  of  which 
does  not  exceed  4  times  the  diameter,  as  in      ~^ 

Fig.  48,  and  under  a  head  of  water  varying  %\|a  c 

from  3  to  20  feet,  give  as  a  mean  value  of  ^  ^ 

the  coetficient  of  efflux,  /<  =  .815,  or  about  ■^/fL^^^^^^ 

^.     Since  the  coefficient  of  efflux  for  a  sim-  ^1=^  ^ 

pie  orifice  in  a  thin  plate  (Art.  94)  is  \i  =  f-^-j 

.615,  it  follows  that,  when  the  other  circum-  i^tt 

stances  are  the  same,  the  discharge  through  ^'9-  ^8 

815 
a  short  cylindrical  tube  =  '-—z  =  1.325  times  the  discharge 

through  a  simple  orifice  in  a  tliin  plate.  These  coefficients 
increase  a  little  when  the  diameter  of  the  tube  becomes 
greater,  and  decrease  a  little  when  the  head  of  water  or  the 
velocity  of  efflux  increases. 

In  this  tube,  the  contraction  of  the  stream  takes  place  at 
the  inlet  db,  and  not  at  the  outlet.  If  a  small  hole  were 
bored  in  the  tube  at  a  or  Z»,  no  water  would  run  out,  but  air 
would  be  sucked  in  ;  and  when  the  hole  is  enlarged,  or  when 
several  of  them  are  made,  the  discharge  with  a  filled  tube 
ceases.  Also,  if  a  tube  be  placed  in  a  vessel  of  water  A,  and 
inserted  in  the  hole  at  h,  the  water  will  rise  in  the  tube  Kb, 
and  run  out  of  the  tube  abed. 

(2)  With  a  compound  mouth-piece,  having     ^ 

an  enlargement  at  its  exterior  orifice  or  out-  vn^'^    <•     ^-^ 

let,  as  well  as  at  its  interior  orifice,  as  in  Fig.  f  ^ 

49,  the  results  of  careful  experiments  f  give  %|6^'^^"^ 

the  coefficient  of  efflux  [i  =  1.5526,  when  the  ^ 
narrow  part  cd  is  treated  as  the  orifice,  thus  f'g-  49 

♦  Experiments  made  by  Michelotti.  t  Made  by  Eytelwein. 


COEFFICIENT  OF  RESIST AXCE.  167 

giving  a  discharge  greater  than  that  which  is  due  to  the  sec- 
tion cd  of  the  pipe.  Since  /fi  =  .615  for  a  simple  orifice,  it 
follows  that  the  discharge  through  the  compound  mouth- 
piece 

1.5536 
=  -'-^Tv-  =  2.5  times  the  discharge  through  a 

simple  orifice  in  a  thin  plate, 

and  =    '      .—  =  1.9  times  the  discharge  through  a 

short  cylindrical  tube. 

In  the  experiments  made  by  Eytelwein,  the  interior  diam- 
eter ah  was  about  1.2  times  the  diameter  cd,  and  the  sides 
ch  and  dk  made  with  each  other  an  angle  of  5°  9'. 

96.  Coefficient  of  Resistance.  —  When  water  flows 
from  a  cistern  through  a  tube  kept  constantly  full,  it  fol- 
lows that  the  coefficient  of  contraction  of  this  mouth-piece 
a  =  unity,  and  that  its  coeflflcient  of  velocity  <f)  =  its  coef- 
ficient of  efl&ux  /u. 

Let  W  be  the  weight  of  water  discharged  with  the  actual 
velocity  v,  and  v^  the  theoretical  velocity  of  discharge  due 
to  the  head  of  water  h.  Then  the  actual  kinetic  energy,  or 
stored  work,  of  the  weight  TF  of  water,  which  issues  with  a 
velocity  v, 

=  ^  W  (Anal.  Mechs.,  Art.  217).  (1) 

But  since  the  theoretical  velocity  of  efl&ux  =  v^,  the 
theoretical  kinetic  energy  or  stored  work  of  the  weight  W 
of  water  discharged 

=  ^W.  (2) 

Hence,  the  loss  of  kinetic  energy  or  stored  work  of  the 
weight  TF  of  water  discharged,  during  the  efflux 

=  (V-^^)|^-  (3) 


168  COEFFICIENT   OF  RESISTANCE. 

But,  from  (1)  of  Art.  93, 

V  =  <pvt',  .-.     v^  =  ^, 

which  ill  (3)  gives 

/I  \  v^ 

stored  work  lost  =  (r^  —  1)  9-  ^«  (4) 

This  loss  of  stored  work  corresponds  to  the  head  of  water 

which  we  can  therefore  consider  as  the  loss  of  head  due  to 

the  resistance  of  efflux,  and  we  can  assume  that,  when  this 

loss  has  been  subtracted,  the  remaining  portion  of  the  head 

is  employed  in  producing  the  velocity. 

The  loss  of  head  in  (5),  which  varies  as  the  square  of  the 

velocity,  is  known  as  the  height  of  resistance. 

1  1^ 

The  coefficient  — ,  —  1,  by  which  the  head  of  water  — 

•^  .  .^ 

must  be  multiplied  in  order  to  obtain  the  height  of  resist- 
ance, i.  e.,  the  ratio  of  the  height  of  resistance  to  the  head 
of  water,  is  called  the  Coefficient  of  Eesistance. 

Cor.  1. — Denote  the  coefficient  of  resistance  by  (3;  then 
we  have 

3  =  i-l,  (6) 

which  in  (5),  and  denoting  the  loss  of  head  or  the  height  of 
resistance  by  z,  we  have 


•2 


'  =  %■  (^) 

CoE.  2. — For  efflux  through  well-formed  smooth  orifices 
in  a  thin  plate,  the  mean  value  ot  (p  =  the  mean  of  .96  and 
.99  (Art.  93),  =  0.975,  and  therefore  we  have,  from  (6), 


COEFFICIENT  OF  RESISTANCE. 


169 


^ = m-  \ 


=  0.052, 


which  in  (4)  gives  for  the  loss  of  energy,  or  stored  work 
lost, 


0.052  ^  W,  or  5.2  per  cent. 
2(/ 


(8) 


Cor.  3. — For  efflux  through  a  short  cylindrical  tube  [Art. 
95,  (1)],  we  have  (p  =  .815,  since  0  =  n,  and  therefore  we 
have,  from  (6), 


P 


\.815/ 


=  0.505, 


which  in  (4)  gives,  for  the  loss  of  energy, 

0.505  ^  W, 
2g 


(9) 


or  nearly  10  times  as  much  as  for  efflux  through  an  orifice 
in  a  thin  plate. 

ScH. — Hence,  if  the  kinetic  energy  of  the  water  is  to  be 
made  use  of,  it  is  better  to  allow  it  to  flow  through  an  ori- 
fice in  a  thin  plate  than  through  a  short  cylindrical  tube. 
But  if  the  edge  of  the  tube  be  rounded  off  where  it  is  united 
to  the  ii\terior  surface  of  the  vessel,  and  shaped  like  the 
contracted  vein,  we  have  /z  =  0  =  .975,  and  the  loss  of 
energy  is  the  same  as  it  is  for  an  orifice  in  a  thin  plate,  i.  e., 
5.2  per  cent. 


Cob.  4. — From  (6)  we  have 


Vi  +^ 


(10) 


which  gives  the  coefficient  of  velocity  in  terms  of  the  coef- 
ficient of  resistance. 


170  RESISTANCE  AND   rUESSURE   OF  FLUIDS. 


EXAMPLE. 

What  is  the  discharge  under  a  head  of  water  of  3  feet 
througli  a  tube  2  inches  in  diameter,  whose  coefficient  of 
resistance  is  jQ  =  0.4  ?     Here  from  (lOj  we  have 

0  =  — L:  =  0.845 ; 
\/l.4 

hence,  from  (1)  of  Art.  93,  we  have,  for  the  actual  veloc- 
ity Vi, 


=  0.845\/64.4x3 

=  0.845x8. 025 a/3  =  11.745  feet; 

h  =  (tV)'^^  =  0.02183  square  feet; 

hence,  the  required  discharge,  from  (1)  of  Art.  94  (since 
«  =  1)  is 

Q  =  kW^h 

=  0.02182  X  11.745  =  0.256  cubic  feet, 

97.  Resistance  and  Pressure  of  Fluids.  —  (1)  By 

the  resistance  of  fluids  is  meant  that  force  by  which  the 
motions  of  bodies  therein  are  impeded.  The  resistance  of  a 
fluid  to  the  motion  of  a  body  is  occasioned  by  the  force 
necessary  to  displace  that  fluid.  Since  the  motion  commu- 
nicated to  a  body  at  rest  by  another  body  impinging  on  it 
with  a  certain  velocity  is  equal  to  the  motion  lost  by  the 
impinging  body,  therefore  the  motion  communicated  to  the 
displaced  fluid  must  be  the  same  as  that  of  the  moving 
body ;  hence  the  work  which  the  fluid  destroys  in  the  mov- 
ing body  will  be  equal  to  the  work  stored  in  the  fluid. 

Let  a  =  the  area  of  the  front  of  the  body  presented  to 
the  fluid,  V  =  the  velocity  of  the  body,  iv  =  the  weight  of 


•    RESISTANCE   AXD   PRESSURE   OF  FLUIDS.  171 

a  cubic  foot  of  the  fluid,  R  =  the  resistance  of  the  fluid  to 
the  motion  of  the  body.     Then, 

weight  of  the  displaced  fluid  per  second  =  avw. 
JJut  tliis  mass  has  a  velocity  of  v  feet  given  to  it; 
.-.     work  generated  per  second  in  displacing  this  fluid 

awv^ 


(1) 


But  this  work  is  performed  by  means  of  a  force  which 
drags  the  body  through  the  water  at  the  rate  of  v  feet  per 
second,  against  an  equal  and  opposite  resistance  R ; 

.'.     Rxv  =  — —  ; 
2g 

"    ^-    2g   '  ^^^ 

that  is,  the  resistance  varies  as  the  square  of  the  ve- 
locity. 

On  account  of  eddies  which  are  formed  round  the  corners 
of  the  body  and  in  the  rear,  the  value  of  R  in  (2)  should  be 
multiplied  by  a  constant  h,  giving 

R  =  kaio^-  (3) 

Rem. — The  constant  k  is  to  be  determined  by  experiment 
for  each  form  of  solid.  For  a  body  whose  transverse  section 
is  circular,  h  does  not  differ  much  from  unity  ;  for  a  flat 
plate  moving  flat-wise,  it  is  about  1.25,  Resistances  of  this 
kind,  however,  are  very  irregular,  and  may  vary  considera- 
bly in  the  course  of  the  same  experiment.  Different  results 
are  therefore  obtained  by  different  experimentalists* 

*  See  Rankine'e  Applied  Mechs.,  p.  598  ;  also  Cotterill's  Applied  Mechs.,  p.  4'i8. 


172    WORK  AND  PRESSURE   OF  A    STREAM   OF   WATER. 

(2)  The  pressure  of  a  current  upon  a  plane  is  equal  to 
the  resistance  suffered  by  the  same  plane  when  moving  in 
the  same  direction  and  with  the  same  velocity  through  the 
fluid;  therefore  (3)  will  also  represent  the  pressure  which 
the  current,  moving  with  the  velocity  v,  would  exert  against 
the  plane  at  rest.     Calling  F  the  pressure,  we  have 

F  =  kaw  ^  •  (4) 

98.  Work  and  Pressure  of  a  Stream  of  Water.— 

To  find  the  ivorh  of  a  stream  of  water  which  impinges 
perpendicularly  upon  the  surface  of  a  heavy  hody 
which  is  itself  in  motion,  and  whose  weight  is  very 
great  as  compared  ivith  that  of  the  impinging  water. 

Let  AB  represent  a  plane  sur-  «, 

face    moving    horizontally    with  HI 

velocity  v^,  while  a  horizontal  jet     ^ — ^_  "^ 
moving  with  greater  velocity  v,  m 

strikes   it  centrally.     Let   W  be  ^''^-  ^°      '  1 

the  weight  of  water  acting  on  the 

surface  per  second.  Then  the  stored  work  9r  kinetic  energy 
of  the  water 

and  if  the  body  were  at  rest,  this  would  be  the  loss  of 
energy. 

From  Anal.  Mechs.,  Art.  208,  (4),  if  m'  be  very  great  as 
compared  with  m,  the  loss  of  kinetic  energy  by  impact 
becomes 

im{v-v,Y.  (2) 

Hence,  if  we  first  suppose  that  the  water  after  impact 
moves  on  with  the  velocity  of  the  body,  we  have  by  (2), 

W 
work  lost  by  impact  =  {v  —  v^)^  —-  (3) 


EXAMPLE.  173 

From  (1)  and  (3)  we  have 

v^  W 

work  done  on  tlie  body  =  —  W —  (v  —  ^1)^^^- 

Now  if  the  water  leaves  the  body,  there  will  be  more  work 
lost,  i.e.,  the  work  reraaming  in  the  water  will  be  lost; 
therefore  we  have 

W      V  ^ 
work  done  on  the  body  =  [v^  —  {v  —  v-^Y]  r -^W 

W 

=  (v-v.Jv^j-  (5) 

Cor.  1. — If  P  denote  the  pressure  of  the  water  against 
the  body,  then  the  work  done  on  tlie  body  =  Pv^,  which 

in  (5)  gives 

W 

P={v-v,)j-  (6) 

If  the  body  is  at  rest,  or"  v^^  =  0,  (6)  becomes 

W 
P  =  —  V.  (7) 

Cor.  2. — Let  a  =  the  section  of  the  pipe,  and  v  =:  the 
velocity  due  to  the  head  of  water  h  ;  then  W  =  62.5av, 
which  in  (7)  gives 

P  =  C>2.5a  X  2h.  (8) 


EXAMPLE. 

To  find  the  work  of  a  stream  of  water  issuing  from  a 
nozzle  with  a  given  velocity. 

Let  V  be  the  given  velocity,  a  the  area  of  the  nozzle,  and 
W  the  weight  of  a  cubic  foot  of  water.     Then  the  weight  of 


174      IMPACT  AGAINST  AXY  SURFACE   OF  REVOLUTION. 


the  water  projected  per  second  =  aiuv,  and  therefore  the 
work  per  second 

that  is,  the  work  varies  as  the  cube  of  the  velocity  of  the 
water. 

Cor. — Let  ^  ^  the  coefficient  of  velocity;  then,  from  (1) 
of  Art.  93,  we  have 


which  in  (1)  gives 

work  per  second  =  (p^awhV^gh. 


(2) 


99.  Impact  of  a  Stream  of  Water  against  any 
Surface  of  Revolution. — Let  BA.C  be  a  surface  of  revo- 
lution, against  which  a  stream  of 
water  FA,  moving  in  the  direction 
of  the  axis  AP  of  the  surface,  im- 
pinges. Let  W  be  tlie  weight  of 
water  discharged  on  the  surface  per 
second,  v  its  velocity,  i\  the  veloc- 
ity of  the  surface,  and  «  the  angle 
BTP  which  the  tangent  HT  to  the 
surface  at  B  makes  with  the  axis 
AP,  or  which  each  filament  HB  of  the  stream  of  water,  on 
leaving  the  surface,  makes  with  the  direction  of  the  axis  BD. 
Then  the  water  irripinges  upon  the  surface  with  the  velocity 
V.  —  Vj  :  and,  if  friction  be  neglected,  the  water  passes  over 
the  surfafce  with  that  velocity,  and  leaves  it  in  a  tangential 
direction,  TH,  TK,  etc.,  with  the  same  velocity.  From 
the  tangential  velocity  BH  =  v  —  v^,  and  the  velocity  BD 
^  Vj  of  the  surface  parallel  to  the  axis,  we  have  the  result- 
ant velocity  BE  =  F  of  the  water,  after  it  has  impinged 
on  the  surface,  by  the  formula  for  the  parallelogram  of 
velocities. 


IMPACT  AGAINST  AXY  SURFACE  OF  REVOLUTION.    175 


V  =  's/(v  —  t^i)^  +  v^^  +  2  (y  —  Vi)  v^  COS  a.       (1) 
Now  the  kinetic  energy  of  the  water  before  impact 

=  .f  w^, 

and  the  kinetic  energy  remaining  in  the  water  after  impact 

=  ^  w;  (2) 

hence,  the  kinetic  energy  transmitted  to  the  surface 

If  P  be  put  for  the  force  or  impulse  against  the  surface, 
then  the  energy  transmitted  to  the  surface  =  Pwj,  which  in 
(3)  gives 

W 

w 

=  [^^  —{V  —  v^y  —  v^^  —  2  {v  —  v^)  v^  cos  «]  — , 

from  (1), 

W 

=  {\-co%a){o  —  v^)v^—;  (4) 

W 
...     p  =  (1  _  cos  «)  {v  -  v^)  —,  ■  (5) 

which  is  the  force  of  the  water  against  the  surface  in  tlie 
direction  of  the  axis. 

That  is,  the  iinpulse  varies  as  the  relative  velocity  of 
the  water. 

Cor.  1. — If  the  surface  moves  with  a  velocity  v^  in  the 
opposite  direction  to  that  of  the  water,  we  have,  from  (5), 

W 

P  =  {l  —  GO&a){v  +  v^)—-  (6) 


176   IMPACT  AGAINST  ANY  SURFACE  OF  REVOLUTION. 


If  the  surface  is  at  rest,  v-^  =  0,  and  (6)  becomes 

W 

P  =  (1  —  cos  a)v  —  - 


(7) 


Cor.  2. — If  a  =  the  area  of  the  cross-section  of  the 
stream,  and  w  =  the  weight  of  a  cubic  foot  of  the  water, 
the  weight  of  the  impinging  water  per  second  is 


W  =  {v  ^  Vj)  aw, 
which  in  (5)  and  (6)  gives 

P  =  (1  —  cos  «)  {v  T  v^y 


and  in  (7)  gives     P  =  (1  —  cos  «)  v^  — 


aw 


(8) 

(9) 
(10) 


That  is,  the  impulse  varies  as  the  square  of  the  rela- 
tive velocity  of  the  water,  and  also  as  the  area  of  the 
cross-section  of  the  stream. 

Cor.  3. — The  impulse  of  the  same  stream  of  water  de- 
pends principally  upon  the  angle  «   at  which   the  water 
moves  off  from  the  axis  after  the  impact. 
If  the  surface  BAC  is  holloAv,  as  in  Fig. 
52,  the  water  after  impact  leaves  the  sur- 
face in  a  direction   opposite  to   that  in 
which  it  strikes  it,  and  thus  much  more 
work  is  done  on  the  body  with  a  surface 
concave  to  the  stream  than  on  one  convex 
to  the  stream,  since  the  work  remaining  in 
the  water  on  leaving  the  former  surface 
will  be  less  than  it  is  in  the  water  on  leaving  the  latter. 
«  =  180°,  we  have  cos  «  =  —  1,  which  in  (5)   and  (6) 
gives 


Fig.  52 


If 


P  =  2  (t)  T  v^) 


W 


(11) 


IMP  A  CT  AGA  INST  ANY  SURFA  CE  OF  RE  VOL  VTION.    1 77 

and  in  (7)  gives  P  =  2v— •  (13) 

Cor.  4. — When  the  surface  is  plane,  as  in  Fig.  50,  a  z= 
90°  and  cos  «  =  0.  Substituting  this  value  in  (5),  (6), 
and  (7),  they  become 

P  =  (.  T  V,)  ^,  (13) 

which  agrees  with  (6)  of  Art.  98 ;  and 

W       v^ 
F  =  V  —  =  —  aic,  from  (8), 
9        ff 


=  2x^  xaw  =  2hx aw ;  (14) 

that  is,  the  normal  impulse  of  water  against  a  plane 
surface  is  equal  to  the  iveight  of  a  column  of  water 
ivhose  base  is  equal  to  the  cross-section  of  the  stream, 
and  whose  height  is  twice  the  head  of  water  to  which 
the  velocity  is  due. 

Cor.  5. — If  the  plane  surface  (Fig.  50)  against  which  the 

stream  impinges  moves  away  with  a  velocity  u  in  a  direction 

which  makes  an  angle  6  with  the  original  direction  of  the 

stream,  the  velocity  of  the  surface  in  the  direction  of  the 

impact  is 

rj  =  M  cos  6, 

which  in  (13)  gives  for  the  impulse, 

p  =  (i,  _  10  cos  d)  --,  (15) 

and  the  work  done  by  it  per  second  is 

W 

Pv^  =  {/'  —  u  cos  6)  u  cos  d  —  (16) 


178 


OBLIQUE  IMPACT. 


Fig.  53 


100.    Oblique    Impact.  —  When    a    stream    injpinges 
obliquely  on  a  plane,  there  are  several  cases,  viz.,  when  the 
water  after  impact  flows  off  in  one, 
two,  or  in  more  directions. 

(1)  Let  the  plane  AB,  upon 
which  the  stream  AC  impinges, 
have  a  border  upon  three  sides  so 
that  the  water  can  flow  off  in  one 

direction  only.  Then  the  impulse  of  the  water  against  the 
surface  in  the  direction  of  the  stream  is,  from  (5)  of  Art.  99, 

P   =    (1   —  cos  «)  {V  —  Vy) 

(2)  Let  the  plane  AB,  upon 
which  the  stream  DC  impinges, 
have  a  border  upon  two  sides  only, 
so  that  the  water  can  flow  off  in 
only  two  directions.  The  stream 
will  divide  itself  into  two  unequal 
parts,  the  greater  part  flowing  off 

in  the  direction  CB,  and  the  other  in  the  direction  CA. 
Let  W^  be  the  weight  of  the  former,  W^  the  weight  of  the 
latter,  and  W  the  whole  weight.  Then  the  total  impulse 
in  the  direction  of  the  stream,  from  (5)  of  Art.  99,  is 

W  W 

/•=(!  —  cos  a)  (y  —  I't)  — -  +  (1  +  cos  «)  {v  —  v{)  — ^ 


Fig.  54 


^  {^^ !!l)  [(1  _  cos  a)  If,  +  (1  +  cos  «)  TFg]. 


(2) 


But  the  conditions  of  equilibrium  of  the  two  portions  of 
the  stream  require  that  the  pressures  on  CB  and  CA  shall 
be  equal  to  each  other  ;  hence 

(1  —  cos  a)  IFj  =  (1  +  cos  a)  W^, 

or,  (1  —  cos  «)  TTj  =  (1  +  cos  a)  {W  —  W^), 


OBLIQUE  IMPACT.  179 

from  which  we  find      PTj  =  |^  (1  +  cos  a)  W, 
and  '  W^  =  i  (1  —  cos  «)  W. 

Substituting  these  values  of  W^  and  TTg  in  (2),  we  have 

which  is  the  total  impulse  in  the  direction  of  the 
stream. 

Dividing  (3)  by  sin  a,  we  obtain 

y  ■y 

CR  =  P  cosec  a  = TTsin  «,  (4) 

which  is  the  normal  impulse. 
Multiplying  (3)  by  cot  «,  we  obtain 

CS  =  P  cot  a  = W  sin  a  cos  a 


g 

V  —  V 


i  Trsin2a,  (5) 


^g 

which  is  the  lateral  impulse. 

Hence,  the  total  impulse  in  the  direction  of  the 
stream  is  propoHional  to  the  square  of  the  sine  of  the 
angle  of  incidence,  the  normal  impulse  to  the  sine  of 
this  angle,  and  the  lateral  impulse  to  the  sine  of 
double  this  angle. 

ScH. — If  the  oblique  plane  has  no  border,  the  water  can 
flow  off  in  all  directions ;  in  this  case  the  impulse  is  in- 
creased, for  «  is  the  smallest  angle  which  the  filaments  of 
water  can  make  with  the  axis,  and  hence  every  filament 
which  does  noit  flow  off  in  the  normal  plane  will  make  with 
the  axis  an  angle  larger  than  «,  and  therefore  from  (3)  will 
exert  a  greater  pressure  than  those  which  do. 


180  MAXIMUM   WORK  DONE  BY  THE  IMPULSE. 

101.   Maximum   Work  done  by  the  Impulse. — 

The  work  done  by  the  impulse  P,  from  (4)  of  Art.  99,  is 

W 
Pv^  —  {1 —  cosa){v —  v^)v^—'  (1) 

The  work  is  zero  when  the  velocity  of  the  surface  v^  =  0, 
and  also  when  it  =  v. 

To  find  the  value  of  Vi  which  makes  this  work  a  maxi- 
mum, we  must  equate  to  zero  its  derivative  with  respect  to 
v^,  which  gives 

V  —  2t'i  =:  0,        or        Vj  =  ^v; 

hence,  the  work  done  by  the  impulse  is  a  maximum 
when  the  surface  moves  in  the  direction  of  the  stream, 
with  half  the  velocity  of  the  stream. 

Substituting  in  (1)  for  i\  its  value,  we  have 

Pv,  =(1-cos«)||tF,  (2) 

which  is  the  maximum  work  done  hy  the  impulse. 

Cor. — If  the  surface  is  a  plane,  as  in  Fig.  50,  a  =  90°, 
and  we  have,  from  (2), 

1  v^ 
P^.^ll^V-  (3) 

That  is,  the  water  transmits  to  the  surface,  in  this 
case,  one-half  of  its  kinetic  energy. 

If  the  surface  is  hollow,  as  in  Fig.  52,  so  that  the  water  is 
reversed,  «  =  180°,  and  we  have,  from  (2), 

Pv.  =  I  JT.  (4) 

In  this  case,  the  water  transmits  to  the  surface  all  of 
its  kinetic  energy.     (See  Weisbach's  Mechs.,  p.  1010.) 


EXAMPLES.  181 


EXAMPLES. 


1.  With  what  velocity  will  water  issue  from  a  small  ori- 
fice 64^  feet  below  the  surface  of  the  liquid  ? 

Ans.  64^  feet. 

2.  If  252  cubic  inclies  of  water  flow  in  one  second  through 
an  opening  of  6  square  inches,  find  the  head  of  water. 

Ans.  2.28  inches. 

3.  If  water  flows  from  a  vessel  whose  cross-section  is  60 
square  inches,  through  a  circular  orifice  in  the  bottom 
5  inclies  in  diameter  under  a  head  of  water  of  24  feet,  find 
its  velocity.  Ans.  41.58. 

4.  A  vessel,  formed  by  the  revolution  of  a  semi-cubical 
parabola  about  its  axis,  which  is  vertical,  is  filled  with  water 
till  the  radius  of  its  surface  is  equal  to  its  height  above  the 
vertex.  Find  the  time  of  emptying  the  vessel  through  a 
small  orifice  at  the  vertex. 

[Let  aif  =:  3^  he  the  equation  of  the  generating  curve, 
and  k  the  area  of  the  orifice.]  Tra'-*     /2^ 

Ans.  ■;rj-\/  — 

5.  A  conical  vessel,  the  radius  of  whose  base  is  r  and  alti- 
tude h,  is  filled  with  water;  the  axis  is  vertical  and  the 
water  issues  through  an  orifice  in  the  vertex,  of  area  k. 
Find  (1)  the  time  in  which  the  surface  of  the  water  mil 
descend  through  one-half  its  altitude,  and  (2)  the  time  in 
which  the  cone  will  empty  itself. 

6.  Find  the  time  in  which  the  cone  in  Ex.  5  would 
empty  itself  through  an  orifice  in  its  base. 

167rr_2     /_A 
15^  V  ^ff 

7.  A  sphere  is  filled  with  water.  Find  the  time  of  empty- 
ing it  through  an  orifice  in  its  bottom.  IQnr^     Ir 


Ans.  -_,-y. 


182  EXAMPLES. 

8.  A  hemisphere  is  filled  with  water.  Find  the  time  of 
emptying  it  (1)  through  an  orifice  in  its  vertex,  and  (2) 
through  an  orifice  in  its  base. 

Ans.    (1)  -^;     (2)  ^^. 

9.  A  rectangular  orifice  is  3  feet  wide  and  1^  feet  high, 
and  the  lower  edge  is  2^  feet  below  the  level  of  the  water. 
Find  the  quantity  discharged  in  1  second. 

A71S.  43.7  cubic  feet. 

10.  An  orifice  in  the  form  of  an  isosceles  triangle,  with 
its  vertex  in  the  surface  of  the  water  has  a  base  of  1  foot 
which  is  horizontal,  and  an  altitude  of  6  inches.  Find  the 
quantity  discharged  in  1  second.     Ans.  1.135  cubic  feet. 

11.  If  the  orifice  in  Ex.  10  has  its  vertex  downwards  and 
its  base  6  inches  below  the  surface  of  the  water,  and  hori- 
zontal, find  the  quantity  discharged  in  1  second. 

A71S.  1.632  cubic  feet. 

12.  If  a  vessel,  when  filled  with  water  to  the  depth  of  4 
feet,  weighs  350  lbs.,  and  if  it  be  drawn  upwards  by  a  weight 
F  of  450  lbs.,  as  in  Fig.  46,  find  the  velocity  of  efflux 
through  an  orifice  in  the  bottom.  Ans.  17.02  feet. 

13.  If  the  vessel  (Fig.  47),  Avhich  is  filled  with  water, 
makes  100  revolutions  per  minute,  and  if  the  orifice  0  is 
2  feet  below  the  surface  of  the  water  at  the  centre,  and  at  a 
distance  of  3  feet  from  the  axis  XX',  find  the  velocity  of 
efflux.  Ans.  33.4  feet. 

14.  Find  the  times  in  which  the  surface  of  water  con- 
tained in  a  vessel,  formed  by  the  revolution  of  the  curve 
y  =:  a^x  about  the  axis  of  x,  will  descend  through  equal 
distances  h,  the  water  issuing  through  a  small  orifice  in  the 

vertex,  and  the  axis  vertical.  na^7i 

Ans.  — -=' 
kV2g 

15.  Water  issues  through  a  small  orifice  16^^  feet  below 
the  surface  of  the  liquid.     If  the  area  of  the  orifice  is  0.1  of 


EXAMPLES.  183 

a  square  foot  and  the  coefficient  of  efflux  is  0.615,  how 
many  cubic  feet  of  water  will  be  discharged  per  minute  ? 

Ans.  118.695. 

16.  A  basin  has  in  it  a  hole  an  inch  square ;  water  in  the 
basin  is  kept  at  a  constant  level  of  9  feet  above  the  hole. 
How  many  cubic  feet  of  water  will  flosv  out  in  1  hour,  the 
coefficient  of  efflux  being  0.6  ?  Ans.  360. 

17.  A  cylindrical  vessel  filled  with  water  is  4  feet  high 
and  1  square  foot  in  cross-section,  and  a  hole  of  1  square 
inch  is  made  in  the  bottom.  If  the  coefficient  of  efflux  is 
0.6,  in  what  time  will  f  of  the  water  be  discharged  ? 

Ans.  60  seconds,  nearly. 

18.  A  cylinder,  the  area  of  whose  cross-section  is  60  sq. 
ft,  is  filled  with  water  to  a  depth  of  12  feet ;  a  small  hole  is 
made  in  its  bottom,  whose  area  is  0.5  square  inches.  In  how 
long  a  time  will  the  depth  of  the  water  be  (1)  8  feet  and  (2) 
4  feet?  Ans.  (1)  45.8  minutes;  (2)  105.4  minutes. 

19.  The  horizontal  section  of  a  cylindrical  vessel  is  100 
square  inches,  its  altitude  is  36  inches,  and  the  area  of  its 
orifice  is  0.1  of  a  square  inch.  If  filled  with  water,  in  what 
time  will  it  empty  itself,  the  coefficient  of  efflux  being 
0.62  ?  Ans.  11  m.  36.5  s. 

20.  What  is  the  discharge  per  second  through  a  rectangu- 
lar orifice  2  feet  wide  and  1  foot  high,  when  the  surface  of 
the  water  is  15  feet  above  the  upper  edge,  the  coefficient  of 
efflux  being  0.611  ?  Ans.  38.6  cubic  feet. 

21.  What  is  the  discharge  per  second  through  a  rectan- 
gular orifice  whose  height  is  8  inches  and  whose  width  is 
2  inches,  under  a  head  of  Avater  of  15  inches  above  the  upper 
edge,  the  coefficient  of  efflux  being  0.628  ? 

Ans.  0.705  cubic  feet. 

22.  If  the  height  of  the  rectangular  orifice  is  15  inches, 
its  width  25  inches,  and  the  head  of  water  is  4^  inches 
above  the  upper  edge,  what  is  the  discharge  per  second,  the 
coefficient  of  efflux  being  0.594  ?      Anf^.   12.19  cubic  feet. 


184  EXAMPLES. 

23.  A  plane  area  moves  perpendicularly  through  water 
in  which  it  is  deeply  imbedded.  Find  the  resistance  per 
square  foot  at  a  speed  of  10  miles  an  hour.   Ans.  269  lbs. 

24.  A  stream  of  water  delivering  100  cubic  feet  per  min- 
ute, at  a  velocity  of  15  feet  per  second,  strikes  an  indefinite 
plane  normally.     Find  the  pressure  on  the  plane. 

Ans.  48.6  lbs. 

25.  If  a  stream  of  water,  the  area  of  whose  cross-section 
is  64  square  inches,  impinges  with  a  velocity  of  40  feet  per 
second  against  the  convex  surface  of  an  immovable  cone,  in 
the  direction  of  its  axis,  the  vertical  angle  of  the  cone 
being  100°,  find  the  impulse.  Ans.  492.16  lbs. 

26.  A  stream  of  water,  the  area  of  whose  cross-section  is 
40  square  inches,  delivers  5  cubic  feet  per  second,  and  strikes 
normally  against  a  plane  surface,  which  moves  away  with  a 
velocity  of  12  feet  per  second.  Find  (1)  the  impulse,  (2) 
the  maximum  work,  and  (3)  the  maximum  impulse. 

Ans.  (1)  58.125  lbs.j  (2)  784.688  ft. -lbs.;  (3)  87.19  lbs. 


CHAPTER    II. 

MOTION  OF  WATER  IN  PIPES  AND  OPEN  CHANNELS. 

102.  Resistance  of  Friction.  —  When  a  thin  plate 
with  sharp  edges,  completely  immersed  in  water,  is  moving 
edgeways  through  the  water,  a  certain  resistance  is  expe- 
rienced, which  must  be  overcome  by  an  external  force. 
This  resistance  acts  along  taugentially  between  the  plate 
and  the  water,  and  so  far  is  analogous  to  the  friction  be- 
tween solid  surfaces,  but  it  follows  quite  different  laws, 
which  have  been  obtained  from  many  observations  and  ex- 
periments, and  which  may  be  stated  as  follows:* 

(1)  The  resistance  of  friction  is  entirely  independent  of 
the  pressure  on  the  surface. 

(2)  It  varies  as  the  area  of  the  surface  in  contact  with 
the  water. 

(3)  It  varies  nearly  as  the  square  of  the  velocity,  f 

Hence,  if  R  be  the  resistance  of  friction,  8  the  area  of  the 
surface,  and  v  the  velocity,  these  laws  may  be  expressed  by 
the  formula, 

R=f8^,  (1) 

where /is  called  the  ^^coefficient  of  friction,"  as  in  the  fric- 
tion of  solid  surfaces.  The  value  of  /  depends  on  the 
smoothness  of  the  surface;  thus,  for  thin  boards,  with  a 
clean,  varnished  surface,  moving  through  water,  it  is  found 

*  Cotterill'8  App.  Mechs.,  p.  458. 

+  At  low  velocities,  of  not  more  than  1  inch  per  second  for  water,  the  resistance 
varies  nearly  as  the  first  power  of  the  velocity.  At  velocities  of  \  foot  per  second, 
and  greater  velocities,  the  resistance  varies  more  nearly  as  the  square  of  the  ve- 
locity. 


186  MOTION  OF  WATER  TN  PIPES. 

to  be  .004,  while  for  a  surface  resembling  medium  sand- 
paper, it  is  .009,  the  units  being  pounds,  feet,  and  seconds.* 

103.  Motion  of  Water  in  Pipes.  — When  water  is 
conveyed  to  any  considerable  distance  in  pipes,  the  friction 
of  the  internal  surface  causes  a  great  resistance  to  the  flow. 
By  the  theoretical  rule,  the  velocity  of  discharge  v  would  be 
due  to  the  vertical  depth  h  through  which  the  water  falls 
(Art.  76) ;  but  owing  to  friction,  theoretical  results  are  of 
very  little  practical  value.  Besides,  the  friction  is  often 
quite  uncertain,  the  central  parts  of  the  stream  move  moi-e 
quickly  than  the  parts  in  immediate  contact  with  the  pipe, 
and,  though  the  circumstances  are  different,  the  velocity 
over  the  internal  surface  is  liable  to  changes,  as  in  the  case 
of  solid  surfaces.  The  value  of  f  therefore  has  to  be  ob- 
tained by  special  experiments,  and  the  results  of  such 
experiments  do  not  always  agree  with  each  other.  It  is 
found,  however,  that/ lies  between  the  limits  .005  and  .01, 
depending  partly  on  the  condition  of  the  internal  surface, 
and  partly  on  the  diameter  and  velocity;  its  value  being 
greater  in  small  pipes  than  in  large  ones,  and  greater  at 
low  velocities  than  at  high  ones.  The  mean  of  these  limits, 
or  .0075,  is  sometimes  taken  for/,  when  there  is  no  special 
cause  for  increased  resistance. 

Let  V  =  the  velocity  of  discharge  in  feet  per  second, 
d  =  the  diameter  of  the  pipe  in  feet,  I  =  the  length  of  the 
pipe  in  feet,  h  =  the  head  or  fall  of  water  in  feet,  and  W 
=  the  weight  of  water  in  pounds  discharged  per  second. 
Let  /'  be  the  resistance  of  friction  due  to  a  unit  of  diam- 
eter, length,  and  velocity  ;  then  the  resistance  in  a  pipe  I 
feet  long  and  d  feet  diameter  with  a  unit  of  velocity  will  be, 
from  (1)  of  Art.  102,  fid;  but  the  quantity  of  water  deliv- 

•  For  large  surfaces,  especially  of  considerable  length,  the  friction  is  very  much 
diminished.  For  instance,  these  values  off  were  obtained  by  experimenting  on  a 
surface  4  feet  long,  moving  10  feet  per  second  :  but  when  the  lentrth  was  20  feet  and 
upwards,  these  values  of/ were  diminished  to  .0025  and  .005  respectively. 


MOTION  OF   WATER   IN  PIPES,  187 

ered  by  this  pipe  will  be  (P  times  that  delivered  by  the  for- 
mer, therefore  for  the  same  quantity  of  water  delivered  as 
by  the  former,  the  resistance  of  friction  in  the  latter  pipe 
will  be 

fid  .,  I 

that  is,  the  resistance  of  friction  in  pipes,  when  the  ve- 
locity is  constant,  varies  directly  as  their  lengths  and 
inversely  as  their  diameters. 

If  we  measure  this  resistance  by  a  column  of  water,  and 
denote  the  height  of  this  column  by  A^,  we  have 

">  =4w  (1) 

where  /  is  a  constant  to  be  determined  by  experiment,  and 
is  called  the  coefficient  of  friction. 

This  height  h^  is  called  the  heigJit  of  resistance  of  friction^ 
which  has  to  be  subtracted  from  the  total  head  li,  in  order 
to  obtain  the  height  necessary  to  produce  the  velocity  v. 
Hence,  the  loss  of  head  or  of  pressure,  in  consequence  of 
the  friction  of  the  water  in  the  pipe,  is  found  by  multiply- 
ing the  head  due  to  the  velocity  by  the  coefficient  f-,  and 

is  greater,  the  greater  the  ratio  of  the  length  to  the  diam- 
eter and  the  greater  the  height  due  to  the  velocity. 

Multiplying  (1)  by  W,  we  obtain  for  the  work  due  to  the 
resistance  of  friction 

that  is,  tlie  loss  of  ivorh  hy  friction  is  the  same  as  that 
of  raising  tJie  water  through  a  height  Aj. 

Cor. — From  (4)  of  Art.  96,  we  have 


188     UNIFORM  FIFE   CONNECTING    TWO   RESERVOIRS. 


loss  of  work  due  to  the  resistance  at  ingress  =  i3  —  W;     (3) 
work  stored  m  the  water  at  discharge  ^  —  W.      (4) 

104.  Uniform  Pipe  connecting  Two  Reservoirs, 
when  all  the  Resistances  are  Considered. — Let  h  be 

the  difference  of  level  of  the  reservoirs,  and  v  the  velocity, 
in  a  pipe  of  length  I  and  diameter  d.    Then  we  have 

work  due  to  the  head  of  water  =:  h  W,  (1) 

which  is  the  whole  work  done  per  second  in  moving  W 
pounds  of  water  from  the  surface  of  one  reservoir  to  the 
surface  of  the  other.  This  work  is  equal  to  the  work  in 
overcoming  all  the  resistances,  together  with  the  work  re- 
maining in  the  water  at  discharge.     That  is,  the  work  is 

expended  in  three  ways:    (1)  The  head  —  ,*  corresponding 

1^ 
to  an  expenditure  of  ^  TF  foot-pounds  of  work,  is  employed 

in  giving  energy  of  motion  to  the  water,  and  is  ultimately 
wasted  in  eddying  motions  in  the  lower  reservoir.     (2)  A 

1^ 
portion  of  head  P  — ,  corresponding  to  an  expenditure  of 

iP 
j3  —  TT  foot-pounds  of  work,  is  employed  in  overcoming  the 

.^ 
resistance  at  the  entrance  to   the  pipe.      (3)   the  head  f 

/-—,  corresponding  to  an  expenditure  of  f  -jk-W  foot- 

pounds  of  work,  is  employed  in  overcoming  the  surface 
friction  of  the  pipe.  Hence,  from  (1),  and  (2),  (3),  (4)  of 
Art.  103,  we  have 

-^  d2g  2g  2g 

•  CaUed  velocity  head.  t  Called/ric<ion  head. 


UNIFORM  PIPE   CONNECTiyO    TWO  RESERVOIRS.    189 

...    A  =  (l+^+4)|,  (2) 

and  V  =  \     '^-^- ^,  .(3) 


or 


^-^^VoWTT^'  W 


where  the  constants  j3  and/ are  to  be  determined  by  experi- 
ment. 

When  V  and  d  are  given,  (2)  is  used  to  determine  h\  when 

li  and  d  are  given,  (4)  is  used  to  determine  v. 

Cor.  1. — If  the  pipe  is  bell-mouthed,  ^  is  about  .08.  If 
the  entrance  to  the  pipe  is  cylindrical,  ^  =  0.505.  Hence, 
1  +  /3  =  1.08  to  1.505.  In  general,  this  is  so  small  com- 
pared with  f-j  that  for  practical  calculations  it  may  be 

neglected ;  i.  e.,  the  losses  of  head,  except  the  loss  in  sur- 
face friction,  are  neglected.  It  is  only  in  short  pipes  and 
at  high  velocities  that  it  is  necessary  to  take  account  of  the 
term  (1  + 13).  For  instance,  in  pipes  for  the  supply  of  tur- 
bines, V  is  usually  limited  to  2  feet  per  second,  and  the  pipe 
is  bell-mouthed.     In  this  case,  we  have 

(1  +  )3)  -^  =  1.08  X  4  X  .0155  =  0.067  foot.* 

In  pipes  for  the  supply  of  towns,  v  may  range  from  2  to 
^\  feet  per  second,  and  then  we  have 

(1  +  /3)  1^  =  0.1  to  0.5  foot. 
In  either  case,  this  amount  of  head  is  small  compared 


190    UNIFORM  FIFE   CONNECTING    TWO  RESERVOIRS. 

with  the  whole  fall  in  the  cases  which  most  commonly 
occur. 

CoE.  2. — For  very  long  pipes,  1 +/J  is  so  small  compared 
with  f-^,  that  it  may  be  neglected  altogether,  and  (2),  (3), 

and  (4)  become 

"  =41'  (^) 

^  =  V7r'  (6) 

V  =  8.025^^.  (7) 

Using  the  value  of/,  as  determined  by  Poncelet,  viz.,/  = 
.02 J,  with  the  value  of  jS  =  .5,  we  have,  from  (3), 


Eytelwein  gave  a  formula  which  nearly  coincides  with 
this.     (See  Storrow  on  Water  Works,  p,  56.) 

When  the  pipe  is  very  long,  d  is  very  small  compared 
with  I,  and  (8)  becomes 

.  =  47.9^^f .  (9) 

Rem. — It  is  immaterial  as  regards  the  velocity,  and  the  quantity 
discharged,  whether  the  pipe  is  horizontal  or  inclined  upwards  or 
downwards,  so  long  as  the  length  of  the  pipe  and  the  total  head,  or 
deptli  of  the  end  of  discliarge  below  the  level  of  the  surface  of  the 
water  in  the  reservoir,  remain  unchanged.  If  the  inclined  pipe  is 
longer  than  the  liorizontal,  of  couree  its  surface  will  present  more  fric- 
tion against  the  motion  of  the  water  than  the  horizontal  one,  and  thus 
diminish  the  velocity  of  discharge  ;  but  if  the  inclined  pipe  be  the 
same  length  as  the  horizontal,  and  have  the  same  head,  then  each  of 
them  will  discharge  the  same  quantity  in  the  same  time. 

It  is  evidently  necessary,  in  every  case,  that  the  entrance  to  the 


COEFFICIENT  OF  FRICTION  FOR  PIPES.  191 

pipe  from  the  reservoir  be  placed  suflBciently  far  below  the  water  sur- 
face of  the  reservoir  to  allow  the  water  to  flow  from  the  reservoir  into 
the  pipe,  as  fast  as  it  afterwards  flows  along  or  through  the  length  ol 
the  pipe  to  the  end  of  discharge.  For  there  must  be  at  least  suflBcient 
head  to  overcome  the  resistance  at  the  entrance  to  the  pipe,  and  to 
allow  the  water  in  the  reservoir  to  flow  out  of  an  opening  freely  into 
the  air  with  that  velocity  which  previous  calculation  shows  it  will 
have  in  the  pipe.  The  remainder  of  the  head,  which  is  employed  in 
overcoming  the  resistance  of  friction,  and  perhaps  other  resistances 
which  will  be  considered  hereafter,  may  be  obtained  by  having  the 
pipe  incline  downwards. 

Since  the  friction  in  pipes  of  the  same  diameter  increases  as  their 
lengths,  when  the  water  first  enters  the  pipe  it  is  opposed  by  but  little 
friction,  and  has  great  velocity ;  but  this  velocity  gradually  diminishes 
as  the  advancing  water  meets  the  friction  along  increased  lengths  of 
the  pipe,  and  finally  becomes  least  when  the  water  fills  the  whole 
length  and  begins  to  flow  from  the  end  of  discharge.  The  velocity 
then  becomes  uniform  along  the  pipe,  and  will  continue  to  be  so,  if  the 
velocity  head  and  head  due  to  the  resistance  at  the  entrance  to  the 
pipe  are  together  sufficient  to  allow  the  water  of  the  reservoir  to  enter 
the  pipe  with  this  same  velocity. 

105.  Coefficient  of  Friction  for  Pipes  Discharg- 
ing Water. — From  the  average  of  a  great  many  experi- 
ments, the  value  of /for  ordinary  pipes  is 

/  =  0.030268.  (1) 

But  practical  experience  shows  that  no  single  value  can 
be  taken  applicable  to  very  different  cases.  The  coeflBcient 
of  friction,  like  the  coefficient  of  efflux,  is  not  perfectly  con- 
stant. It  is  greater  for  low  velocities  than  for  high  ones, 
t.  e.,  the  resistance  of  friction  of  the  water  in  pipes  does  not 
increase  exactly  as  the  square  of  the  velocity.  Prony  and 
Eytelwein  assumed  that  the  loss  of  head  by  the  resistance 
of  friction  increases  with  the  first  power  of  the  velocity  and 
with  its  square ;  and  hence  they  established  for  this  loss  of 
head  the  formula 


192  COEFFICIENT  OF  FRICTION  FOR   PIPES. 

in  which  «j  and  a^  denote  constants  to  be  determined  by 
experiment. 

In  order  to  determine  these  constants,  these  authors 
availed  themselves  of  51  experiments  made  at  different  times 
by  Couplet,  Bossut,  and  du  Buat  upon  the  flow  of  water 
through  long  pipes.  From  these  51  experiments,  the  fol- 
lowing numerical  values  were  obtained : 

Prony  obtained,     «i  =  0.0000693,        a^  =  0.0013932. 
Eytelwein,  a^  z=  0.0000894,        a^  =  0.0011213. 

D'Aubuisson,*       «i  =  0.0000753,        a^  =  0.0013700. 

Taking  the  value  of  hi,  and  substituting  it  in  (2)  of  Art. 
104,  instead  of  the  value  of  h^  as  given  in  (1)  of  Art.  103, 
we  have 

^  =  (l+^)|  +  («i^'  +  «2^)^-  (3) 

1+3 
Putting  — ^ —  =  5,  and  reducing,  (3)  becomes 

hd  =  Mv^  +  cc^h^  -h  (tjv,  (4) 

from  which  the  value  of  v  may  be  found.  But  the  follow- 
ing method  of  approximating  to  the  value  of  v  is  more 
convenient.     From  (4)  we  have 


^'^  +  mt^i^^'=  y 


hd 


hd  +  ad 


Expanding  the  first  member  by  the  binomial  theorem, 
and  neglecting  all  the  terms  of  the  expansion  after  the  sec- 
ond, since  a^  is  considerably  greater  than  a^,  we  have 


.fi  . -A 11-./ 


lid  _^ 
hd  +  a^V 


•  Weisbach's  Mechs.,  p.  866. 


COEFFICIENT  OF  FRICTION  FOR  PIPES.  193 


/        hd  €Cy^l  >   . 

•'•    ^  ~  V  MTVt^l  ~  2{bd-{-  a^l)  ^' 

Now  if  the  pipe  is  cylindrical,  (i  =  0.505,  from  Cor,  3 
of  Art.  96,  and  therefore  we  have 

_  1  +i3  _  L505 

=  .0234, 

and  taking   «i  =  .00007   and   a^  =  .00042,*  and  substi- 
tuting these  values  in  (5)  and  reducing,  we  have 


_       /  2380M I . 

^  ~   y  I  +  5^d       12(1 +  5^)'  ^' 

CoK. — When  h  is  not  very  small,  the  last  term  of  (6)  may 
be  neglected,  and  we  have 

/2d80hd  ,„. 

-  =  VrT-5T^'  (^^ 

which  is  very  nearly  the  same  as  (8)  of  Art.  104. 

Wlien  the  pipe  is  very  long,  d  is  very  small  compared 
with  I,  and  (6)  becomes 


v^sj 


2S80hd       1  ,^, 

1 la*  (^> 


When  d  is  expressed  in  inches  and  all  the  other  dimen- 
sions in  feet,  (8)  of  Art.  104  becomes 

V  =  .  /I9l5l.  (9) 

ScH. — The  following  short  table  gives  Weisbach's  values 
of  the  coeflBcient  of  friction  for  diiferent  velocities  in  feet 
-♦^r  second :  f 

♦  Tate's  Mech.  Phil.,  p.  293. 

t  Ency.  Brit.,  Art.  Hydromechanics. 


194         THE   QUANTITY  DISCHARGED   FROM  PIPES. 


V   = 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

f  — 
J   — 

.0686 

.0527 

.0457 

.0415 

.0387 

.0365 

.0349 

.0336 

.0325 

V   = 

1 

li 

H 

2 

3 

4 

6 

8 

12 

/= 

.0315 

.0297 

.0284 

.0265 

.0243 

.0230 

.0214 

.0205 

.0193 

EXAMPLE 

The  length  of  a  water-pipe  is  5780  feet,  the  head  of  water 
is  170  feet,  and  the  diameter  of  the  pipe  is  6  inches.  Re- 
quired the  velocity  of  discharge. 


2380  X  170  X  .5 


5780 


By  (6),  we  have 

/ 
^  ~  V  5780 +54X.5        12(5780  +  54  X. 5) 

By  (8),  we  have 

By  (8)  of  Art.  104,  we  have 

V: 


=  5.81. 


2380  X  170  X. 5  _  J:_  _  ^  j^o 

5780  12  "" 


47.9^1 


170  X. 5 


5780  +  54X.5 

By  (9)  of  Ai-t.  104,  we  have 

^170"xT5 
5780 


=  5.8  feet,  nearly. 


V  =  47.9y - 


=  5.8. 


It  will  be  observed  that  these  results  are  very  nearly  the 
same. 

106.  The  Quantity  Discharged  from  Pipes.— Let 
Q  be  the  number  of  cubic  feet  discharged  per  second;  then 
Q  is  given  by  the  formula 


THE  QUANTITY  DISCHARGED  FROM  PIPES.         195 

^  =  ^^y  =  0.7854d?2v;  (1) 

and  on  substituting  the  value  of  v  obtained  from  (1)  of  Art. 
103,  this  becomes 


Q 


which  gives  the  value  of  Q  in  cubic  feet  per  second,  since 
all  the  dimensions  are  in  feet. 

If  we  require  the  number  of  gallons  discharged  per  min- 
ute for  a  diameter  of  d  inches,  (1)  becomes 

Q  =  cJ^-d\  (3) 

wrhere  C  is  a  constant  whose  value  for  /  =  .03  is  30,  but 
which  is  often  taken  somewhat  less  (say  27),  to  allow  for 
contingencies.* 

Assuming  that  1  cubic  foot  =  6.2322  gallons,  we  have, 
from  (1),  for  the  number  of  gallons  discharged  in  24  hours, 

O  =  -^-  d?v  X  86400  X  6.2322 

570 

=  2936.86^2^.  (4) 

From  (1),  we  have 

.  =  if,  =  1.2733 1,  (5) 

which,  in  (1)  of  Art.  103,  gives 

that  is,  the  height  of  resistance  of  friction  in  pipes 
varies  inversely  as  the  fifth  power  of  the  diameters, 
and  directly  as  the  length  of  the  pipes. 


*  See  Cotterill's  App.  Mecbe.,  p.  468. 


196  EXAMPLES. 

Hence,  if  we  wish  to  conduct  a  given  quantity  of  water 
through  a  pipe  with  as  little  loss  of  head  as  possible,  we 
must  make  the  pipe  as  short  and  its  diameter  as  large  as 
we  can.  If  the  diameter  of  one  pipe  is  double  that  of 
another,  the  friction  in  the  former  is  -5V  of  that  in  the 
latter. 

Cor.— Putting  !-{■&  =  1.505,  and  —  =  0.0155,  we 
have,  from  (2)  and  (4)  of  Art.  104,  ^ 

7i  =  /l.505 +/.-)  0.0155^2,  (7) 

and  V  =  8.025^ 


1.505  +/.^ 

EXAMPLES. 

1.  How  many  gallons  of  water  would  the  pipe  in  the  ex- 
ample of  Art.  105  deliver  in  24  hours  ? 

Here  i;  =  5.8  and  (Z  =  6  inches ;  we  have,  from  (4), 

Q  =  2930.86x62x5.8 

=  613216  gallons  in  24  hours. 

2.  What  must  the  head  of  water  be,  when  a  set  of  pipes 
150  feet  long  and  5  inches  in  diameter  is  required  to  deliver 
85  cubic  feet  of  water  per  minute  ? 

Here  we  have,  from  (5), 

25     122 

V  =  1.2732  X  J^  X  -4-  =  3.056  feet, 

60       a* 

and  therefore  (Art.  105,  Sch.),  /  =  .0243,  which  in  (7) 
gives 

h  =  (1.505  +  .0243  X  ^^^^~^\  .0155  X  3.056' 
=  (1.505  +  8.748)  .0155  X  9.339  =  1.484  feet. 


THE  DIAMETER    OF  PIPES.  197 

3.  Solve  Ex.    1   by   using   the  value   of  v  as  obtained 
from  (8). 

From  (8),  we  have 


V  z=  S.OSOA  / 

^  1.505  +  / 


505  +/5^ 

•^    .5 

Since  v  is  somewhere  between  3  and  10,  Ave  assume  /  = 
,03,  and  obtain 


V  =  8.025a/- 


170 


.505  -1-  231.20 
=  6.859. 

But  V  =  6.9  gives  more  correctly  (Art.  105,  Sch.)  /  = 
.021,  and  therefore  we  have 


V  =  8.025a/— 


170 


.505  +  244.265 
=  6.695, 
which  gives  the  true  value  to  the  first  decimal  place. 

The  discharge,  from  (4),  is 

Q  =  2936.86  x  62  x  6.7 

=  708370.632  gallons  in  24  hours. 

This  result  is  somewhat  larger  than  that  obtained  from 
the  value  of  v  in  (8)  of  Art.  104. 

107.  The  Diameter  of  Pipes.— Substituting  in  (1) 
of  Art.  106  the  value  of  v  given  in  (9)  of  Art.  105,  we  have 


«  =  5?6V 


191.2M  ^ 


198  EXAMPLE. 

Or,  by  logarithms, 

\ogd  =  -^  [2. 2450532  +  3 log  ^ -f  log  (Z 4- 4. 5^)— log ^],    (2) 

where  d  is  in  inches,  and  all  the  other  terms  are  in  feet. 

When  the  pipes  are  very  long,  or  when  d  is  small  as  com- 
pared with  /,  (2)  becomes 

\ogd  =  \  (2.2450532  +  2  log  (?  +  log  ?  —  log  h).    (3) 

Eem. — The  value  of  d  can  be  obtained  from  (1)  only  by 
successive  approximations.  When  considerable  accuracy  is 
required,  find  the  value  of  d  from  (3),  and  substitute  it  in 
(2),  which  will  give  a  first  approximate  value  of  d ;  and 
this  again  substituted  in  (2)  will  give  a  closer  approximate 
value ;  and  so  on  to  any  required  degree  of  accuracy.  Gen- 
erally the  first  approximate  value  will  be  found  sufficiently 
accurate  for  all  practical  purposes. 

EXAMPLE. 

What  is  the  diameter  of  a  pipe  which  shall  deliver  25000 
gallons  of  water  per  hour,  when  the  length  of  the  pipe  is 
2500  feet,  and  the  head  of  water  225  feet  ? 

Here  h  =  225  and  I  =  2500,  and  the  number  of  cubic 
feet  delivered  per  second  is 

„  25000  ,  ,,  .-         , 

^  =  60^^ir6:2322  =  ^-^^^^  "^"^^y- 

Substituting  in  (3),  we  have 
\ogd  =  ^  (2.2450532  +  2  log  1.1143  +  log  2500  —  log  225) 

=  -^(1.9870309  +  3.3979400)  =  .67696; 
;.    d  =  4.7530  inches. 


SUDDEN  ENLARGEMENT  OF  SECTION.  199 

Substituting  this  value  of  d  in  (2),  we  have 

log  d  =  -\  [1.9870309  +  log  (2500  +  4.5  x  4.7533)] 

=  .6777337; 

.*.     d  =z  4.761  inches, 

which  approximation  is  suflBciently  accurate  for  all  practi- 
cal purposes. 

108.   Sudden  Enlargement  of  Section. — Whenever 
there  is  a  change  in  tliC  cross-section  of  a  pipe  or  any  other 
conduit,  there  is  a  change  of  velocity,  the  velocity  being 
inversely  proportional  to  the  cross-section  of  the  stream 
(Art.  75).      If  the  cross-section   of  a 
pipe  is  suddenly  changed,  there  is  a 
sudden  change  in  the  velocity  of  the 
current  of  water,  and  therefore  there  is 
a  loss  of  kinetic  energy.     Thus,  sup- 
pose the  pipe  AECF  is  suddenly  en- 
larged in  section  at  BD ;  then,  as  the  f '9-  55 
water  in  the  smaller  pipe  has  a  greater 
velocity  than  the  water  in  the  larger  one,  there  will  be  an 
abrupt  change  of  velocity  at  BD,  and  this  change  of  veloc- 
ity will  be  accompanied  by  a  loss  of  kinetic  energy,  in  the 
same  way  as  when  two  inelastic  bodies  impinge  upon  each 
other. 

Let  V  and  v'  be  the  velocities  of  the  water  in  the  smaller 
and  larger  pipes,  respectively,  a  and  a'  the  areas  of  the  sec- 
tions of  these  pipes,  W  the  weight  of  water  discharged  from 
ABCD  per  second,  and  W  the  whole  weight  in  BEFD. 

Now  as  the  water  moves  out  of  the  smaller  pipe  into  the 
larger  one,  it  impinges  against  the  more  slowly  moving  cur- 
rent in  that  pipe,  and  after  the  impact  the  two  bodies  of 
water,  IF  and  W,  move  on  together  with  the  common  ve- 
locity v'.  And  since  in  this  case  W  is  very  small  compared 
with  W,  we  have,  from  (3)  of  Art.  98, 


200  SUDDEN  ENLARGEMENT  OF  SECTION. 

work  lost  by  the  water  at  the  abrupt  change  of  velocity 

=  ("-"71-  (1) 

If  7i^  is  the  head  of  water  corresponding  to  this  loss  of 
work,  we  have,  for  the  work  lost, 

A,Tf=(^-t.7^;  (2) 

and  therefore  the  head  lost  is 

K  =  '^-  (3) 

Hence,  the  head  lost  at  the  abrupt  change  of  velocity 
is  measured  hy  the  height  due  to  this  change  of  ve- 
locity. 

Since  we  have  v  '.  v'  i:  a'  '.  a, 

a    , 
a 

substituting  this  in  (3),  we  have,  for  the  loss  of  head, 

/a'         \2 
where  we  put  ^  ~  \ ^) '  (^) 

which  is  the  corresponding  coefficient  of  resistance.* 

ScH. — When  the  edges  are  rounded  off  so  as  to  cause  a 
gradual  passage  from  one  pipe  into  the  other,  and  the  dif- 
ference in  the  pipes  is  small,  the  loss  of  work  as  shown  by 
experiment  is  very  small. 

*  First  found  by  Borda.    (See  Weisbacb's  Hechs.,  p.  884.) 


SUDDEN  CONTRACTION  OF  SECTION. 


201 


109.  Sudden  Coutraction  of  Section. — When  water 
passes  from  a  larger  to  a  smaller  section,  as  in  Figs.  56,  57, 
where  it  passes  from  the  pipe  AB  into 
the  narrower  pipe  CEDF,  a  contraction 
is  formed,  and  the  contracted  stream 
abruptly  expands  to  fill  the  section  of 
the  pipe,  thereby  causing  a  loss  of  head 
by  this  sudden  enlargement,  precisely  as 
in  Art.  108. 

(1)  Ijet  a  be  the  area  of  the  section 
and  V  the  velocity  of  the  stream  at  EF.     Then,  if  c  is  the 
coefficient  of  contraction,  the  section  of  the  stream  at  CD 
will  be  ca,  and  the  velocity  v'  at  this  section  will  be  found 
by  means  of  the  formula 


V  ca 


vax 


V    =  -' 

c 


Hence,  the  loss  of  head  in  passing  from  CD  to  EF  is 


_  (t;'  —  vY  _  /I  _  .V]^ 


'^9 


if  j3  is  put  for  ( 1  j  • 


If  c  is  taken  =  0.64  (Art.  92),  we  have 
which  in  (1)  gives        h^  =  0.316  —• 


(1) 


(2) 


ScH.  1. — The  value  of  the  coefficient  of  contraction  in 
this  case  is,  however,  not  well  ascertained,  and  the  result  is 


202  SUDBEN  CONTRACTIOX  OF  SECTION. 

somewhat  modified  by  friction.  For  water  entering  a  cylin- 
drical pipe  from  a  reservoir  of  indefinitely  large  size,  experi- 
ment shows  that  j3  is  increased  by  the  resistance  at  the 
entrance  into  the  pipe,  and  by  the  friction  of  the  water  in 
the  pipe,  to  0.505,  so  that  (1)  becomes 

Ji^  =  0.505^.  (3) 

(2)  If  there  is  a  diaphragm  at  the 
mouth  of  the  pipe,  as  at  AB,  Fig.  57, 
with  an  opening  ab,  whose  cross-section 
is  smaller  than  the  cross-section  of  the 
pipe  CEDF,  let  a'  be  the  area  of  this  ori-  ''•g-  57 

fice.     Then  if  c  is  the  coeflBcient  of  con- 
traction as  before,  the  area  of  the  contracted  stream  is  ca', 
and  the  velocity  v'  at  the  contracted  section  will  be 

a 

V   =  —,v; 
ca 

hence,  the  head  lost  in  passing  from  the  contracted  section 
to  the  pipe  at  EF  is 


^^-        2a        ~  \ca'       V 


2g  \ca  1 2g 

where  the  corresponding  coefficient  of  resistance  is 

ScH.  2. — Weisbach  has  found  experimentally  the  follow- 
ing values  of  the  coefficients  c  and  i3,  when  the  stream  ap- 
proaching the  orifice,  as  in  Fig.  57,  was  considerably  larger 
than  the  orifice.* 

*  Ency.  Brit.,  Vol.  Xni.,  Art.  Hydromechaniat. 


EXAMPLE. 


203 


a'  _ 
a 

0.1 

0.2 

0.3 

0.4 

0.5 

.616 

.614 

.612 

.610 

.607 

(3  = 

231.7 

50.99 

19.78 

9.612 

5.256 

a'  _ 
a 

0.6 

0.7 

0.8 

0.9 

1.0 

c  = 

.605 

.603 

.601 

.598 

.596 

(i  = 

3.077 

1.876 

1.169 

0.734 

0.480 

ScH.  3. — When  the  edges  are  rounded  off  so  the  contrac- 
tion is  very  gradual,  the  loss  of  work  is  very  much  dimin- 
ished. 


EXAMPLE, 


What  is  the  discharge  through  the  orifice  in  Fig.  57, 
when  the  head  is  1^  feet,  the  diameter  of  the  contracted 
circular  ori^ce  =  1\  inches,  and  that  of  the  pipe  CEDF  = 
2  inches  ? 


Here  we  have 


i = m = ^ = -' 


and  therefore  from  the  table  c  =■  0.606,  and  from  (5),* 


"  =  (9^  -  'J=  '■ 


Hence,  the  whole  head  is 


74. 


i  =  ^  +  ;„  =  (i  +  (3)|: 


*  Since  c  comes  between  two  numbers  in  the  table,  0  is  fonnd  more  accurately 
from  (5). 


204  ELBOWS. 

therefore,  the  velocity  of  efflux  is 

VWi         8.025  VTs 

V  =  -  .-- 1—  =  T=^=—  =  4.51, 

^/l+  a  a/4.74 

and  consequently  the  discharge,  from  (1)  of  Art.  106,  is 

Q  =  ~cPv  =  "^x  4x4.51x12 
4  4 

=  170  cubic  inches. 

110.  Elbows.  —  When  pipes  are  bent  so  as  to  form 
elbows,  they  present  resistances  to  the  motion  of  water  in 
them  ;  and  these  resistances,  like  many  other  phenomena  of 
efflux,  can  be  determined  only 
by  experiment.  If  a  pipe  ACB 
forms  an  elbow,  the  stream  sep- 
arates itself  from  the  inner  sur- 
face of  the  second  branch  of  the 
pipe,  in  consequence  of  the  cen- 
trifugal force,  if  the  second 
branch  is  very  short,  termi- 
nating, for  instance,  at  ab,  the  efflux  will  be  smaller  than 
the  full  cross-section  of  the  pipe.  But  if  the  second  branch 
is  longer,  terminating  at  B,  an  eddy  is  formed  at  D,  and 
beyond  this  the  pipe  is  again  filled,  so  that  the  velocity  of 
efflux  V  is  less  than  the  velocity  at  D.  This  diminution  of 
the  velocity  of  efflux  must  be  treated  in  the  same  way  as 
the  resistance  produced  by  a  contraction  in  the  pipe 
(Art.  109). 

Hence,  if  a  is  the  cross-section  of  the  pipe,  and  c  is  the 
coefficient  of  contraction,  the  section  of  the  stream  at  D  is 
ca,  and  the  velocity  v'  of  the  contracted  stream  is 


Fig.  58 


ELBOWS.  206 

and  hence  the  loss  of  head  in  passing  from  D  to  B  is 
_  {v'  —  vY  _  {I         \2«;2 

The  coefficient  of  contraction  c,  and  therefore  the  cor- 
responding coefficient  of  resistance  P,  depends  upon  the 
angle  of  deviation  BCE.  From  experiments  with  a  pipe  1;|- 
inches  in  diameter,  Weisbach  found  the  coefficient  of  re- 
sistance to  be 

13  =  0.9457  sin2 1  +  2.047  sin^f ,  (2) 

by  which  he  computed  a  series  of  coefficients  of  resistance 
for  different  angles  of  deviation.*  From  (2)  it  follows  that 
the  kinetic  energy  of  water  in  pipes  is  considerably  dimin- 
ished by  elbows.  If  the  elbow  is  right-angled,  we  have, 
from  (2),  |3  =  0.9846,  which  in  (1)  gives 


h.  =  0.9846 -; 
2g' 

lieiice,  at  a  right-angled  elbow,  the  loss  of  head  is  nearly 
equal  to  the  head  due  to  the  velocity. 

ScH. — If  to  one  elbow  ACB  another  elbow  is  joined,  the 
second  one  turning  the  stream  to  the  same  side  as  the  first 
one,  there  is  no  further  contraction  of  the  stream,  and 
therefore,  for  efflux  with  full  cross-section,  (3  is  no  larger 
than  for  a  single  elbow.  But  if  the  second  elbow  turns  the 
stream  to  the  opposite  side,  the  contraction  is  a  double  one, 
and  the  coefficient  of  resistance  is  consequently  twice  as 
great  as  for  a  single  elbow. 

*  Bncy.  Brit.,  Vol.  XIL,  p.  487. 


206 


BENDS. 


111.  Bends. — When  the  pipes  have  curved  bends,  the 
resistance  is  much  less  than  in  elbows.  If  a  pipe  ACB  is 
curved,  it  also,  in  consequence  of  the 
centrifugal  force,  causes  the  stream  to 
separate  itself  from  the  concave  surface, 
and  to  form  a  partial  contraction.  If 
the  bend  terminates  at  BD,  the  cross- 
section  of  the  stream  at  its  outlet  is 
smaller  than  that  of  the  pipe.  But  if 
the  bend  is  terminated  by  a  long  straight 
pipe  BF,  an  eddy  is  formed  at  D,  and  beyond  this  the  pipe 
is  again  filled,  so  that  the  velocity  of  efflux  v  is  less  than  the 
velocity  at  D. 

If  c  is  the  coefficient  of  contraction,  the  velocity  v'  of  the 
contracted  stream  is 

"  ='c' 
and  hence  the  loss  of  head  in  passing  from  D  to  F  is 
(v'  —  vY        (I  \2y2 


'^g 


^9 


'%■ 


(1) 


This  is  Weisbach's  method,  but  the  coefficient  of  contrac- 
tion for  bends  is  not  very  satisfactorily  ascertained 

If  r  =  the  radius  of  the  pipe  =  MH  =  HC,  and  p  = 
the  radius  of  curvature  =  HO,  then  Weisbach's  formula 
for  the  coefficient  of  resistance  at  a  bend  in  a  pipe  of  circu- 
lar section  is 


j3  =  0.131  -f  1.847  (~)^; 


and  for  bends  with  rectangular  cross-sections, 
|3  =  0.124  +  3.104  (|-)", 


(3) 


(3) 


EQUIVALENT  PIPES.  20"? 

where  s  is  the  length  of  the  side  of  the  section  parallel  to 
the  radius  of  curvature  p.     (See  Weisbach's  Mechs.,  p.  897.) 

11  Ir/.  Pipe  of  Uniform  Diameter  Equivalent  to 
one  of  Yarying  Diameter.— Pipes  for  the  supply  of 
towns  *  often  consist  of  a  series  of  lengths,  the  diameter  for 
each  length  being  the  same,  but  differing  from  those  of  the 
other  lengths.  In  approximate  calculations  of  the  head 
lost  in  such  pipes,  it  is  generally  accurate  enough  to  neglect 
the  smaller  losses  of  head  and  to  regard  only  the  friction  of 
the  pipe,  and  then  the  calculations  may  be  facilitated  by 
reducing  the  pipe  to  one  of  uniform  diameter,  having  the 
same  loss  of  head.  Such  a  uniform  pipe  is  called  an  equiv- 
alent pipe. 


Let  A  be  the  pipe  of  varia-  ^ 

ble    diameter,    and    B    the  i,  ^^  ~ 

equivalent  pipe  of  uniform  — 

diameter.     In   A  let  Zj,  Zg,  = 

etc.,  be  the  lengths,  d^,  d^,  Fig.  59a 

etc.,   the  diameters,   v■^^,  v^, 

the  velocities  for  the  successive  portions,  and  let  /,  d,  v,  be 
the  corresponding  quantities  for  the  equivalent  uniform 
pipe.    Then  the  total  loss  of  head  in  A  due  to  friction  is 

and  in  the  uniform  pipe  B, 

h        /-^^ 
''  d'Zg 

If  these  pipes  are  equivalent,  we  have 

/•li^'^/A^V/ii^'  +  etc.  (1) 


*  Snch  pipes  are  called  water  mains. 


208  DISCHARGE  DIMINISHING    UNIFORMLY. 

But  since  the  discharge  is  the  same  for  all  portions, 
4  4     ^  4     ^ 

••     ^^"^^'^"2'     ^'^'^^rf-a^     etc.  (2) 

Then  supposing  that  /  is  constant  for  all  the  pipes,  we 
have,  from  (1)  and  (2), 

which  gives  the  length  of  the  equivalent  uniform  pipe 
which  would  have  the  same  total  loss  of  head,  for  any  given 
discharge,  as  the  pipe  of  varying  diameter. 

Cor. — If  the  lengths  of  the  successive  portions  are  all 
equal,  we  have  Z^  =  Zg  =  /g  =  etc.,  and  (3)  becomes 


111&.  Pipe  of  Uniform  Diameter  with  Discharge 
Diminishing  Uniformly  along  its  Length.  — In  the 

case  of  a  branch  main,  the 

water  is  delivered  at  nearly      A^ ,     •         ,        P       ^  ,c        B 

equal    distances    to   service  , ,  ,  p= 

pipes  along  the  route.     Let  Pig  59^ 

AB  be  a  main  of  diameter  d 

and  length  L;  let  Q^  cubic  feet  per  second  enter  at  A,  and 

let  q  cubic  feet  per  foot  of  its  length  be  delivered  to  service 

pipes.     Then  at  any  point  C,  I  feet  from  A,  the  discharge  is 

Q  =  Qo  —  ql'     Consider  a  short  length  dl  at  P.     The  loss 

of  head  in  that  length  is 


GENERAL  FORMULA  FOR  ALL  THE  RESISTANCES.   209 

--^{Q,-qlfdl 


Hence,  the  whole  head  lost  in  the  length  AB  is 

or,  putting  P  ■=  qL,  the  total  discharge  through  the  ser- 
vice pipes  between  A  and  B,  (1)  becomes 

The  discharge  at  tlie  end  B  of  the  pipe  is  Q^  —  P.  If 
the  pipe  is  so  long  that  Q^  —  P  =  0,  all  the  water  passes 
into  the  service  pipes,  and  (2)  becomes 

"  =  11^,^-  (^) 

(See  Ency.  Brit.,  Vol.  XII.,  p.  486.) 

112.  General  Formula  when  all  the  Resistances 
to  the  Flow  of  Water  are  Considered. — Let  (i^  be 

the  coefficient  of  resistance  for  enlargements  and  contrac- 
tions (Arts.  108  and  109),  and  /Jg  the  coefficient  of  resist- 
ance for  elbows  and  bends  (Arts.  110  and  111).  Then 
adding  together  (3)  of  Art.  105,  (4)  of  Arts.  108  and  109, 
(1)  of  Arts.  110  and  111,  we  have  for  the  entire  head  Ti, 

A=(l+/3)|+K.  +  «,.^)^  +  3.|  +  (3,| 
=  («,t>+a,t>2)j+(l+|3  +  /3,  +(3,)^,         (1) 


210  EXAMPLE. 

where  the  values  of  a^  and  a^  are  given  in  Art.  105,  /S^  in 
Art.  109,  iSg  in  Art.  110,  and  d  =  .505  to  .08. 

Neglecting  a^,  since  it  is  very  small  compared  with  a^ 
(Art.  105),  and  putting  /  =  '^ga^  =  .03,  from  (1)  of  Art. 
105,  we  have,  from  (1), 

^  =  (4  +  l+/5  +  /3,  +^3)|-  (2) 

ScH.— An  enlargement  should  be  made  in  the  pipe  at  any 
considerable  bondings  ;  and  when  any  change  takes  place  in 
the  diameter  of  the  pipe,  the  parts  at  the  junction  should 
be  rounded  off.  At  all  considerable  bends,  where  the  pipe 
changes  from  ascending  to  descending,  a  provision  should 
be  made  for  clearing  the  pipe  of  the  air  which  is  disengaged 
from  the  water.  Unless  some  provision  is  made  for  the 
escape  of  this  air,  it  will  accumulate  in  the  highest  bends 
and  obstruct  the  flow  of  the  current. 


EXAMPLE. 

In  the  example  of  Art.  105  there  are  40  bends  in  the 
pipe,  each  having  the  radius  of  curvature  exceeding  ten 
times  the  radius  of  the  pipe.    Find  the  velocity  of  eflSux. 

Here  /  =  .03,    /3  =  .505,     (3^=0, 

/Jg  =  .131  +  lM7{^y^  =  .1316; 

.-.     40/32  =  5.263,     I  =  5780,     d  =  .5,     h  =  170. 

Hence,  from  (2),  we  have 


170  =  (.03  x^  +  1.505  +  5.263) 


64^ 
=  353.568 


64i' 
V  =  5.562  feet  per  second 


FLOW  OF  WATER   IX  RIVERS  AND    CANALS.         211 

For  practical  calculations  on  the  flow  of  water  in  pipes, 
see  Ency.  Brit,  Vol.  XII.,  p.  488. 

113.   Flow   of  Water  ill    Rivers    and    Canals.— 

When  water  flows  in  a  pipe,  the  section  at  any  point  is  de- 
termined by  the  form  of  the  boundary.  When  it  flows  in 
an  open  channel  with  free  upper  surface,  the  section  de- 
pends on  the  velocity  due  to  the  kinetic  conditions.  The 
bottom  of  the  channel  and  the  two  banks  are  called  the  bed 
of  the  stream.  A  section  of  the  stream  at  right  angles  to 
the  direction  in  which  it  is  flowing  is  called  a  transverse 
section,  and  of  the  line  bounding  this  section,  the  part  that 
is  beneath  the  water  surface  is  called  the  wetted  perimeter. 
A  vertical  section  in  the  direction  of  the  stream  is  called  the 
loiif/itudinal  section  oxj)rofile. 

Let  ABCD  represent  a  longitudinal 
section  of  a  limited  portion  of  a  stream, 
AD,  BC,  two  transverse  sections,  AB 
the  surface  of  the  stream,  DC  the  bot 
tom  of  the  channel,  and  AE  a  horizon-  ^'S'  *° 

tal  line.     Let  I  =  the  length  of  AB  in 
feet ;  h  =  BE,  the  difference  of  level  of  the  water  surface 
in  feet  at  the  two  extremities  of  the  distance  I;  6  =  the 

angle  BAE,  the  slope  of  the  stream ;  sin  0  ^  -  =r  the  sine 

of  the  slope,  or  the  fall  of  the  water  surface  in  one  foot ; 
a  =  the  area  of  the  transverse  section  at  BC  in  square  feet ; 
p  =  the  length  of  the  wetted  perimeter  of  the  transverse 

section  at  BC  ;   r  =  - ,  the  hydraulic  mean  depth,  or  the 

mean  radius  of  the  section  ;    Q  =  the  discharge  through 

the  section  at  BC  in  cubic  feet  per  second  ;  v  =2  -  =  the 

a 

mean  velocity  of  the  stream  in  feet  per  second,  which  is 

taken  as  the  common  velocity  of  all  the  particles. 


212      DIFFERENT    VELOCITIES  IN  A    CROSS-SECTION. 

114.  Different  Yelocities  in  a  Cross-Section.— The 

velocity  of  the  water  is  not  uniform  in  all  points  of  the 
same  transverse  section.  In  all  actual  streams  the  different 
fluid  filaments  have  different  velocities.  The  adhesion  of 
the  water  to  the  bed  of  the  channel,  and  the  cohesion  of  the 
molecules  of  water  cause  the  particles  of  water  nearest  to 
the  sides  and  bed  of  the  channel  to  be  most  hindered  in 
their  motion.  For  this  reason,  the  velocity  is  much  less  at 
the  bottom  and  sides  than  it  is  at  the  surface  and  centre. 
According  to  some  authors,  the  maximum  velocity  in  a 
straight  river  is  generally  found  in  the  middle  of  its  sur- 
face, or  in  that  part  of  the  surface  where  the  water  is  the 
deepest.*  Theoretically  we  should  expe.ct  thi3,  but  practi- 
cally it  is  often  very  different. 

The  theory  adopted  by  most  modern  writers  is  the  fol- 
lowing: The  motion  of  the  water  being  caused  solely  by  the 
slope  of  the  surface,  the  velocity  in  all  parts  of  any  trans- 
verse section  of  the  river  would  be  equal,  were  it  not  for  the 
retarding  influence  of  the  bed.  The  layer  of  elementary 
particles  next  to  the  bed  adheres  firmly  to  it  by  virtue  of 
the  force  of  adhesion.  The  next  layer  is  retarded  partly  by 
the  cohesion  existing  between  it  and  the  first,  partly  by  the 
friction,  and  partly  by  the  loss  of  kinetic  energy  arising  from 
constant  collision  with  the  irregularities  which  correspond 
to  those  of  the  bed.  The  next  layer  is  retarded  in  the  same 
manner,  but  in  a  less  degree.  Thus,  according  to  this 
theory,  the  effect  of  the  resistances  is  diminished  as  the 
distance  from  the  bed  is  increased  ;  and  assuming,  as  is 
usually  done,  that  no  sensible  resistance  is  experienced  from 
the  air,  the  maximum  velocity  should  be  found  in  the  sur- 
face filament  situated  at  the  greatest  distance  from  the  bed. 
The  many  experiments,  however,  which  have  been  made  to 
determine  the  actual  variation  in  velocity  at  different 
depths,  and  upon  the  surface,  at  different  distances  from  the 
banks,  give  very  different  results. 

*  Welsbach's  Mechs.,  p.  956  ;  also  Tate's  Mech.  Phil.,  p.  808. 


DtFFHRENT   VELOCITIES  IN  A    CROSS-SECTION.      313 

Focacci  found  that  in  a  canal  5  feet  deep,  the  maximum  velocity 
was  from  2  to  3.5  feet  below  the  surface. 

Defontaine  states  that  in  calm  weather  the  velocity  of  the  Rhine  is 
greatest  at  the  surface. 

Raucourt  made  experiments  upon  the  Neva  where  it  is  900  feet  wide 
and  of  regular  section,  the  maximum  depth  being  63  feet.  When  the 
river  was  frozen  over,  the  maximum  velocity  (2  feet  7  inches  per  sec- 
ond) was  found  a  little  below  the  middle  of  the  deepest  vertical,  where 
it  was  nearly  double  the  velocity  at  the  surface  and  bottom,  which 
were  nearly  equal  to  each  other.  In  summer,  he  found  the  maximum 
velocity  was  near  the  surface  in  calm  weather ;  but  when  a  strong 
wind  was  blowing  up  stream,  the  surface  velocity  was  greatly  dimin- 
ished, so  that  it  hardly  exceeded  that  at  the  bottom.  He  considers  the 
law  of  diminution  of  velocity  to  be  given  by  the  ordinates  of  an  ellipse 
whose  vertex  is  a  little  below  the  bottom,  and  whose  minor  axis  is  a 
little  below  the  surface. 

Hennocque  found  the  maximum  velocity  in  the  Rhine  to  be,  in  calm 
weather,  or  with  a  light  wind,  \  of  the  depth  below  the  surface ;  in  a 
strong  wind  up  stream,  it  was  a  little  below  mid-depth  ;  in  a  strong 
wind  down  stream,  it  was  at  the  surface. 

Baumgarten  found  in  the  Garonne  that  the  maximum  velocity  was 
generally  at  the  surface,  but  that  in  one  section  (about  325  feet  wide) 
it  was  always  below  the  surface. 

D'Aubuisson  considers  that  the  velocity  diminishes  slowly  at  first, 
as  the  depth  increases,  but  that  near  the  bottom  it  is  more  rapid.  The 
bottom  velocity,  however,  is  always  more  than  half  that  of  the  surface. 
Boileau  found,  by  experiment  in  a  small  canal,  that  the  maximum 
velocity  was  ]  to  1  of  the  depth  below  the  surface.  Below  this  point, 
the  velocity  diminished  rapidly,  and  nearly  in  the  ratio  of  the  ordinates 
of  the  parabola  whose  axis  was  at  the  surface.  He  decided,  from  a 
discussion  of  the  experiments  of  Defontaine,  Hennocque,  and  Baum- 
garten, that  in  large  rivers  the  maximum  velocity  is  by  no  means  al- 
ways at  the  surface.* 

It  will  be  seen  from  this  synopsis  that  there  is  a  great 
diversity  among  the  results  obtained  by  different  experi- 
menters, and  that  no  mathematical  relation,  of  sufficiently 
general  application  to  constitute  a  practical  law,  has  been 
yet  discovered. 

♦  See  Beport  on  the  Hydraulics  of  the  MissisBippi  Biver,  by  Humphreys  and 
Abbott,  pp.  200,  etc. 


214      DIFFERENT   VELOCITIES  IN  A    CROSS-SECTION. 

The  velocities  observed  on  any  given  longitudinal  section,  at  any 
given  moment,  do  not  form,  when  plotted,  any  regular  curve.  But  if 
a  series  of  observations  are  taken  at  each  depth,  and  the  results  aver- 
aged, the  mean  velocities  at  each  depth,  when  plotted,  give  a  regular 
curve  agreeing  very  fairly  with  a  parabola  whose  axis  is  horizontal,  cor- 
responding to  the  position  of  the  filament  of  maximum  velocity.  All 
the  best  observations  show  that  the  maximum  velocity  is  to  be  found 
at  some  distance  below  the  free  surface. 

In  the  experiments  on  the  Mississippi  River,  the  velocities  on  any 
longitudinal  section,  in  calm  weather,  were  found  to  be  represented 
very  fairly  by  a  ])arabola,  the  greatest  velocity  being  at  -f'^  of  the  depth 
of  the  stream  from  the  surface.  With  a  vrind  blowing  down  stream, 
the  surface  velocity  is  increased  and  the  axis  of  tlie  parabola  approaches 
the  surface.  With  a  wind  blowing  up  stream,  the  surface  velocity  is 
diminished  and  the  axis  of  the  parabola  is  lowered,  sometimes  to  half 
the  depth  of  the  stream.  The  observers  on  the  Mississippi  drew  from 
their  observations  the  conclusion  that  there  was  an  energetic  retarding 
action  at  the  surface  of  a  stream,  like  that  at  the  bottom  and  sides.  If 
there  were  such  a  retarding  action,  the  position  of  the  filament  of  max- 
imum velocity  below  the  surface  would  be  explained.  If  there  were 
no  such  resistance,  the  maximum  velocity  should  be  at  the  sur- 
face. 

It  is  not  difficult  to  understand  that  a  wind,  acting  on  surface  rip- 
ples, should  accelerate  or  retard  the  surface  motion  of  the  stream,  and 
the  Mississippi  results  may  be  accepted  so  far  as  showing  that  the  sur- 
face velocity  of  a  stream  is  variable  when  the  mean  velocity  of  the 
stream  is  constant.  Hence  observations  of  surface  velocity,  by  floats 
or  othermse,  should  only  be  made  in  very  calm  weather.  But  it  is 
verj'  difficult  to  suppose  that,  in  still  air,  there  is  a  resistance  at  the 
free  surface  of  the  stream  at  all  analogous  to  that  at  the  sides  and 
bottom.  In  very  careful  experiments,  Boileau  found  the  maximum 
velocity,  though  raised  a  little  above  its  position  for  calm  weather, 
still  at  a  considerable  distance  below  tbe  surface,  even  when  the  vnnd 
was  blowing  down  stream  with  a  velocity  greater  than  that  of  the 
stream,  and  when  the  action  of  the  air  must  have  been  an  accelerating 
and  not  a  retarding  action.  Prof.  James  Thomson  has  given  a  much 
more  probable  explanation  of  the  diminution  of  the  velocity  at  and 
near  the  free  surface.  He  points  out  that  portions  of  water,  with  a 
diminished  velocity  from  retardation  by  the  sides  or  bottom,  are  thrown 
off  in  eddying  masses  and  mingle  with  the  rest  of  the  stream.  These 
eddying  masses  modify  the  velocity  in  all  parts  of  the  stream,  but 
have  their  greatest  influence  at  the  free  surface.     Reaching  the  free 


TRANSVERSE  SECTION  OF  THE  STREAM.  215 

surface,  they  spread  out  and  remain  there,  mingling  with  the  water  at 
that  level,  and  diminishing  the  velocity  which  would  otherwise  be 
found  there.* 

115.   Transverse   Section   of  the  Stream.  — The 

form  of  tlie  transverse  section  and  the  direction  of  the  cur- 
rent have  such  an  effect  upon  the  velocity  at  the  surface,  at 
different  distances  from  the  banks,  that  there  can  be  no 
definite  law  of  change.  There  is  generally  an  increase  of 
velocity,  as  the  distance  from  the  banks  is  increased,  until 
the  maximum  point  is  reached.  That  portion  of  the  river 
where  the  water  has  its  maximum  velocity  is  called  the  line 
of  current  or  axis  of  the  stream,  and  the  deepest  portion  of 
the  stream  is  called  the  mid-channel.  When  the  stream 
bends,  its  axis  is  generally  near  the  concave  shore. 

It  is  observed  that  the  surface  of  a  stream,  in  any  cross- 
section,  is  highest  where  the  velocity  is  greatest,  which  is 
accounted  for  by  the  fact  that,  when  the  water  is  in  motion, 
it  exerts  less  pressure  at  right  angles  to  the  direction  of  its 
motion  than  when  it  is  at  rest,  and  therefore,  where  the 
velocity  is  greatest  the  water  must  be  highest,  to  balance 
the  pressure  at  the  sides,  where  the  velocity  is  less. 

It  frequently  happens  that,  while  the  mass  of  the  water 
in  a  river  is  flowing  on  down  the  river,  the  water  next  the 
shore  is  running  up  the  river.  It  is  no  unusual  thing  to 
find  a  swift  current  and  a  corresponding  fall  on  one  shore 
doivn  stream,  and  on  the  opposite  shore  a  visible  current 
and  an  appreciable  fall  up  stream;  i.  e.,  on  one  side  of  the 
river  the  water  is  often  running  rapidly  up  stream,  while  on 
the  other  side  it  is  running  with  equal  or  greater  rapidity 
down  stream.  The  apparent  slope  at  every  point  is  affected 
by  the  bends  of  the  river,  and  by  the  centrifugal  force 
acquired  by  the  water  in  sweeping  round  the  curves,  and  by 
the  eddies  which  form  on  the  opposite  side.  The  surface  of 
the  river  is  not  therefore  a  plane,  but  a  complicated  warped 

*  Ency.  Brit.,  Vol,  XII.,  p.  497. 


216   RATIO  OF  MEAN  TO  GREATEST  SURFACE  VELOCITY. 

surface,  varying  from  point  to  point,  and  inclining  alter- 
nately from  side  to  side.* 

116.  Mean  Velocity. — The  mean  velocity  of  the  water 
in  a  cross-section  is  equal  to  the  quotient  arising  from 
dividing  the  discharge  per  second  by  the  area  of  the  trans- 
verse section. 

When  the  discharge  per  second  is  not  known,  the  mean 
velocity  may  be  determined  by  measuring  the  velocities  in 
all  parts  of  the  transverse  section,  and  taking  a  mean  of  the 
results.  If  the  transvei-se  section  is  irregular  in  form,  the 
only  accurate  manner  of  determining  the  mean  velocity  is 
to  divide  this  section  into  partial  areas  so  small  that  the 
velocity  tluoughout  each  may  be  considered  invariable.  The 
discharge  is  tiien  equal  to  the  sum  of  the  products  of  these 
partial  areas  by  their  velocities. 

Let  a^,  «2,  «3,  etc.,  be  the  small  partial  areas  into  which 
the  transverse  section  is  divided,  and  v^,  v^,  v^,  etc.,  the 
velocities  in  these  small  areas.     Then  the  whole  area  is 

tf  =  «j  4-  «3  4-  flg  +  etc.,  (1) 

and  the  whole  discharge  is 

av  =  a^v^  +  a^v^  -}-  a^v^  +  etc.;  (2) 

therefore  the  mean  velocity  is 

^  _  a^Vi  +  ggt'g  +  a^v^  -t-  etc. 

«1    +  «2    +  ^3   +  6tC.  ^    ^ 

117.  Ratio  of  Mean  to  Greatest  Surface  Velocity. 

— It  is  often  very  important  to  be  able  to  deduce  the  mean 
velocity  from  observation  of  the  greatest  surface  velocity. 
The  greatest  surface  velocity  may  be  determined  by  floats. 
Unfortunately,  however,  the  ratio  of  the  maximum  surface 
velocity  to  the  mean  velocity  is  extremely  variable  ;  and  it 

♦  See  Report  on  the  MissJssippi, 


RATIO  OP  MEAN  TO  GREATEST  SURFACE  VELOCITY.   21? 

has  formed  the  subject  of  much  careful  investigation.  Put- 
ting Vq  for  the  greatest  surface  velocity,  and  Vm  for  the 
mean  velocity  of  the   whole   cross-section,   the   following 

values  have  been  found  for    *" 


V 


De  Prony,  experiments  on  small  wooden  channels,  0.8164 

Experiments  on  the  Seine, 0.62 

Destrem  and  De  Prony,  experiments  on  the  Neva,  0. 78 

Boileau,  experiments  on  canals, 0.82 

Baumgarten,  experiments  on  the  Garonne,    .    .    .  0.80 

Brunings  (mean), 0.85 

Cunningham,  Solani  aqueduct,  0.823 

Dubuat,  experiments  on  small  canals  (mean),    .    .  0.83 
Dupuit,   from   theoretical  considerations,  believes 
the  ratio  to  varv  between  0.67  and  1.00. 


Various  formuhe  have  been  proposed  for  determining  the 
tio  —  •     Bazin  four 
empirical  expression. 


ratio  —     Bazin  found  from  his  experiments  the  following 

^0 


Vm  =  Vo  -  25A\/rd,  (1) 

where  r  is  the  hydraulic  mean  depth,  and  6  the  slope  of  the 
stream  (Art.  113). 

Prony  found  the  following  formula, 

*''"  -      V,  +  10.33"  ^^^ 

The  ratio  of  the  mean  velocity  to  the  surface  velocity  in 
one  longitudinal  section  is  better  ascertained  than  the  ratio 
of  the  greatest  surface  velocity  to  the  mean  velocity  of  the 
whole  cross-section.  Let  the  river  be  divided  into  a  num- 
ber of  compartments  by  equidistant  longitudinal  planes,  and 
the  surface  velocity  be  observed  in  each  compartment ;  then 
from  this  the  mean  velocity  in  each  compartment  and  the 


218  PROCESSES  FOR    GAUGING   STREAMS. 

discharge  can  be  computed.  The  sum  of  the  partial  dis- 
charges will  be  the  total  discharge  of  the  stream.  The  fol- 
lowing formula*  is  convenient  for  determining  the  ratio  of 
the  surface  velocity  to  the  mean  velocity  in  the  same  verti- 
cal. Let  V  be  the  mean  and  V  the  surface  velocity  in  any 
given  vertical  longitudinal  section,  the  depth  of  which  is  h. 

v_  _  1  +  0.1 478  a/^ 
^  ~  1  +  0.2216\/a' 

ScH. — In  the  gaugings  of  the  Mississippi,  it  was  found 
that  the  mid-depth  velocity  differed  by  only  a  very  small 
quantity  from  the  mean  velocity  in  the  vertical  section,  and 
it  was  uninfluenced  by  wind.  If  therefore  a  series  of  mid- 
depth  velocities  are  determined,  they  may  be  taken  to  be 
the  mean  velocities  of  the  compartments  in  which  they 
occur,  and  no  formula  of  reduction  is  necessary. 

118.  Processes  for  Gauging  Streams. — The  dis- 
charge of  large  creeks,  canals,  and  rivers,  can  be  measured 
only  by  means  of  hydrometers,  which  are  instruments  for 
indicating  the  velocity.  The  simplest  of  these  instruments 
are  surf  ace  floats  ;  these  are  convenient  for  determining  the 
surface  velocities  of  a  stream,  though  their  use  is  difficult 
near  the  banks.  Any  floating  body  can  be  used  for  this 
purpose  ;  but  it  is  safer  to  employ  bodies  of  medium  size, 
and  of  but  little  less  specific  gravity  than  the  water  itself. 
Very  large  bodies  do  not  easily  assume  the  velocity  of  the 
water,  and  very  small  bodies,  especially  when  they  project 
much  above  the  surface  of  thfe  water,  are  easily  disturbed  in 
their  motion  by  accidental  circumstances,  such  as  wind,  etc. 

The  floats  may  be  small  balls  of  wood,  of  wax,  or  of  hol- 
low metal,  so  loaded  as  to  float  nearly  flush  with  the  water 
surface.  To  make  them  visible,  they  may  have  a  vertical 
painted  stem.     In  experiments  on  the  Seine,  cork  balls  1| 

♦  Given  by  Eyner  in  Erbkam's  Zeitschrift  for  1875. 


PROCESSES  FOR    GAUGING   STREAMS.  219 

inches  diameter  were  used,  loaded  to  float  flush  with  the 
water  surface,  and  provided  with  a  stem.  Bits  of  solid 
wood,  and  bottles  filled  with  water  until  nearly  submerged, 
have  often  been  used  for  surface  floats.  Boileau  proposes 
balls  of  soft  wax,  on  account  of  their  adhesive  properties. 
In  Captain  Cunningham's  observations,  the  floats  were  thin 
circular  disks  of  English  deal,  3  inches  diameter  and  \  inch 
thick.  For  observations  near  the  banks,  floats  1  inch  diam- 
eter and  \  inch  thick  were  used.  To  render  them  visible,  a 
tuft  of  cotton  wool  was  used,  loosely  fixed  in  a  hole  at  the 
centre. 

The  velocity  is  obtained  by  allowing  the  float  to  be  car- 
ried down,  and  noting  the  time  of  passage  over  a  measured 
length  of  the  stream.  If  t  is  the  time  in  which  the  float 
passes  over  a  lengtli  /,  which  has  been  previously  measured, 

and  staked  off  on  the  shore,  then  the  velocity  v  is  v  =  —. 

To  mark  out  distinctly  the  length  of  stream  over  which  the 
floats  pass,  two  ropes  may  be  stretched  across  the  stream  at 
a  distance  apart,  which  varies  usually  from  50  to  250  feet, 
according  to  the  size  and  rapidity  of  the  river.  To  mark 
the  precise  position  at  which  the  floats  cross  the  ropes,  Capt. 
Cunningham,  in  his  experiments,  used  short  white  rope 
pendants,  hanging  so  as  nearly  to  touch  the  water.  In  this 
case  the  streams  were  80  to  180  feet  wide. 

In  wider  streams  the  use  of  ropes  to           C  p 

mark  the  length  of  run  is  impossible  ;  in     p|^  i'Z~_ 
such  cases,  recourse  must  be  had  to  some     M ^ 


such  method  as  the  following:  Let  AB  /^  B 

be  the  measured  length  :=  I,  on  one  side  p.    g, 

of  the  river.  Put  two  rods  C  and  D,  by 
means  of  a  suitable  instrument,  in  such  a  position  upon  the 
other  side  of  the  river  that  the  lines  CA  and  DB  shall  be 
perpendicular  to  AB.  Then  the  observer,  placed  behind  A, 
notes  by  his  watch  tlie  instant  the  float  E,  which  has  been 
placed  in  the  water  some  distance  above,  arrives  at  the  line 


PROCESSES  FOR    GAUGING   STREAMS. 

AC,  aud  then,  passing  down  to  B,  lie  observes  the  instant 
that  the  float  arrives  at  the  line  BD.  By  subtracting  the 
time  of  the  first  observation  from  that  of  the  second,  he 
obtains  the  time  t  in  which  the  space  I  is  described. 

For  measuring  the  velocity  below  the  surface,  double 
floats  *  are  used.  They  are  of  various  kinds,  usually  con- 
sisting of  small  surface  floats,  supporting  by  cords  larger 
submerged  bodies.  Suppose  two  equal  and  similar  floats, 
connected  by  a  string,  wire,  or  thin  wire  chain.  Let  one 
float  be  a  little  heavier,  and  the  other  a  little  lighter  than 
water,  so  that  only  a  small  portion  of  the  latter  will  project 
above  the  surface  of  the  water.  We  first  determine  by  a 
single  float  the  surface  velocity  Vg;  we  then  determine  the 
Telocity  of  the  connected  floats,  which  will  be  the  mean  of 
the  surface  velocity  and  the  velocity  at  the  depth  at  which 
the  heavier  float  swims.  If  I'd  is  the  velocity  at  the  depth 
to  which  the  lower  float  sinks,  we  have,  calling  v  the  mean 
velocity, 

'  =  -!-' 

.'.     I'a  =  2v  —  Vg.  (1) 

By  connecting  the  floats  successively  by  longer  and  longer 
pieces  of  wire,  we  obtain  in  this  way  the  velocities  at  greater 
and  greater  depths. 

To  obtain  the  mean  velocity  in  a  perpendicular,  a  floating 
staff  or  rod  is  often  employed.  This  consists  of  a  cylindrical 
rod,  loaded  at  the  lower  end  so  as  to  float  nearly  vertical  in 
water.  A  wooden  rod,  with  a  metal  cap  at  the  bottom  in 
which  shot  can  be  placed,  so  as  to  prevent  more  than  the 
head  from  projecting  above  the  water  surface,  answers  well, 
and  sometimes  the  wooden  rod  is  made  of  short  pieces  which 
can  be  screwed  together  so  as  to  suit  streams  of  different 
depths.     A  tuft  of  cotton  wool  at  the  top  serves  to  make 

♦  First  used  by  da  VincL 


MOST  ECONOMICAL  FORM  OF  TRANSVERSE  SECTION.   221 

the  float  more  easily  visible.  Such  a  rod,  so  adjusted  in 
length  that  it  sinks  nearly  to  the  bed  of  the  stream,  gives 
directly  the  mean  velocity  of  the  whole  vertical  section  in 
which  it  floats.  (For  a  complete  description  of  gauging 
streams,  see  ''  Report  on  the  Mississippi.*') 

119.  Most  Economical  Form  of  Transverse  Sec- 
tion.— The  best  form  of  the  transverse  section  must  be  that 
which  presents  the  least  resistance  to  a  given  quantity  of 
water  flowing  through  the  channel.  From  Art.  103,  the 
resistance  of  the  bed  of  the  stream,  in  consequence  of  the 
adhesion  and  friction,  varies  directly  as  the  surface  of  con- 
tact, and  consequently  as  the  wetted  perimeter  j!>  (Art.  113), 
and  inversely  as  the  area  of  tlie  transverse  section,  {.  e.,  the 

P 
resistance  of  the  bed  of  the  stream  varies  as  -•     In  order, 

a 

therefore,  to  have  the  least  resistance  from  friction,  the 
form  of  the  section  must  be  that  which  has  the  least  perim- 
eter for  a  given  area,  i.  e.,  the  wetted  perimeter^;  must  be  a 
minimum  for  a  given  area  a,  or  the  area  must  be  a  maximum 
for  a  given  wetted  perimeter.  Now,  among  all  figures  of 
the  same  number  of  sides,  the  regular  one,  and  among  all 
the  regular  ones,  the  one  with  the  greatest  number  of  sides 
has  the  smallest  perimeter  for  a  given  area.  Hence,  for 
closed  pipes,  the  resistance  of  friction  is  the  smallest  when 
the  transverse  section  is  a  circle  ;  but  in  open  chaunels,  the 
upper  surface,  being  free, 


CA 


or  in  contact  with  the  air 
alone,  must  not  be  in- 
cluded in  the  perimeter. 

A  horizontal  line  DC,      a   (^)    b 
passing  through  the  centre 
of  the  square  AF,  divides 

the  area  and  perimeter  into  two  equal  parts,  and  what  has 
been  said  of  the  square  is  true  of  these  halves ;  hence,  of  all 
rectangular  forms  of  transverse  sections,  the  half  sqnare 


222     TRAPEZOIDAL   SECTION  OF  LEAST  RESTSTAKCE. 

ABCD  is  the  one  which  causes  the  least  resistance  of  fric- 
tion, and  therefore  is  the  best  for  open  cliannels.  Also,  of 
all  trapezoidal  sections,  the  semi-hexagon  ABCD  is  the  one 
which  causes  the  least  resistance  of  friction  ;  and  so  on  to 
the  other  cases.  But  the  semicircle  will  present  less  resist- 
ance of  friction  than  the  semi-hexagon,  and  this  latter  less 
than  the  semi-square.  The  half  decagon  offers  still  less 
resistance  than  the  half  hexagon  or  the  half  square.  The 
circular  and  square  sections  are  used  only  for  troughs 
made  of  iron,  stone,  or  wood.  The  trapezoid  is  employed  in 
«ianals,  which  are  dug  out  or  walled  up.  It  is  very  rare  that 
other  forms  are  used,  owing  to  the  diflBculty  of  constructing 
them. 

120.    Trapezoidal   Section   of  a  Canal  of  Least 
Resistance,  when  the  Slope  of  the  Sides  is  Given. 

— Let  ABCD  be  the  section.     Put 

X  =  AB,  the  width  of  the  bottom, 

y  =  BE,  the  depth,  and  6  =  BCE, 

the  angle  of  the  slope,  which  is  to 

be  considered  as  a  given   quantity, 

dependent  upon  the  nature  of  the 

ground  in  which  the  canal  is  excavated,  and  a  =  the  given 

area  of  the  section  ABCD. 

Tlien  the  wetted  perimeter  of  the  section  is 

p  =  AB  -{-  2BC 
=  X  +  2y  cosec  0.  (1) 

'  I'he  area  of  the  section  is 

a  =:  xy  -\-  y^  cot  6 ; 

a  —  «/2  cot  B  f^. 

•••    ^  =  -^ »  (^) 

which  in  (1)  gives 

a  —  y^  cot  6  ,„v 

p  = -^ \-  2y  cosec  6.  (d) 


EXAMPLE.  223 

To  find  the  value  of  y  which  makes  this  a  minimum,  we 
must  equate  to  zero  its  derivative  with  respect  to  y,  which 
gives 

«sin6>_^ 

^        2  -  cos  0 '  ^*' 


r~a  sin  0  ,. 

•••  y  =  y^^r^oJe-  <^> 

Hence,  for  a  given  angle  of  slope  6,  and  for  a  given  area 
fl,  the  trapezoidal  seciion  of  least  resistance  is  detennined  by 
(2)  and  (5). 

Consequently,  the  width  CD  of  the  top  is 

Qjy  =  X  -\-2y  cot  e 


a 


=  -  +  ycote;  (6) 

P 
and  the  value  of  — ,  from  (3),  is    * 

a  ^ 

p        1       2  —  cos  6 

—  =  -  -i ^—7^  V 

a        y         a  sin  9    "^ 

=  ^  [from  (4)].  (7) 

EXAMPLE. 

What  dimensions  should  be  given  to  the  transverse  sec- 
tion of  a  canal,  when  the  angle  of  slope  of  its  banks  is  to  be 
40°,  and  when  it  is  to  carry  75  cubic  feet  of  water  with  a 
mean  velocity  of  3  feet  ? 

Here  we  have 

a  =^  ^  =  25  square  feet; 

and  hence,  from  (5),  we  have  the  depth 


/  25  sin  40°         ^     /0.64279        ,  ^aq  ^    x 

=  V  2-=:-^4()^  =  V  1:2339-6  =  ^-^^^  ^^^^- 


^24  UNIFORM  MOTION. 

From  (2),  we  have  the  width  at  the  bottom, 

25 
X  =  — --  —  3.609  cotan  40° 
o.d09 

=  6.927  —  4.301  =  2.626  feet. 

The  width  of  the  top,  from  (6),  is 

CD  =r  2.626  +  7.218  cot  40°  =  11.228  feet 

The  wetted  perimeter  is 

■^  sin  6 

7  21 8 
—  2.626  +    ■     ,^o  =  13.855  feet ; 
sm  40 

and  the  ratio  which  determines  the  resistance  of  friction  is 

^  =  -  =  0.5542. 
a        y 

Rem. — In  a  transverse  section  in  the  shape  of  the  half  of 
a  regular  hexagon,  where  d  =  60°,  a:  =  4.39,  y  =  3.80, 
width  CD  =  8.78,  and  p  =  13.17  feet,  we  have,  for  the 
resistance  of  friction, 

f  =  %"  =  «■-. 

which  is  less  than  that  found  for  the  above  trapezoid. 

121.  Uniform  Motion. — When  water  flows  in  an  open 
channel,  the  velocity  continues  to  increase  so  long  as  the 
accelerating  force  exceeds  the  resisting  force  of  friction ; 
but  when  these  forces  are  equal  to  each  other,  the  velocity 
of  the  stream  becomes  uniform.  When  the  velocity  is  uni- 
form, the  entire  head  h  is  employed  in  overcoming  the  fric- 
tion upon  the  bed.    Therefore,  the  height  of  the  column  of 


UNIFORM  MOTION.  225 

water  due  to  the  resistance  of  friction  must  be  equal  to  the 
fall     The  height  due  to  the  resistance  of  friction  increases 

with  — ,  with  the  length  I,  and  with  the  square  of  the  ve- 
locity V  (Art.  102).     Hence,  from  (1)  of  Art.  103,  we  have 

''=A$,  (I) 

in  which  /  is  an  empirical  number,  which  is  called  the  coef- 
ficient of  friction. 
Solving  (1)  for  v,  we  have 


hlqha 


(3) 


According  to  Eytelwein's  reduction  of  the  ninety-one  ob- 
servations and  experiments  made  by  du  Buat,  Briinings, 
Funk,  and  Woltmann,  /  =  0.007565,  which  in  (1)  gives 

/*  =  0.007565^.^.  (3) 

a  ^g  ^  ' 

Kwe  put  g  =  32.2  feet,  (2)  and  (3)  become 

"  =  ''Wji'  (*> 

h  =  0.00011747  ^  v^,  (5) 

For  the  number  of  cubic  feet  of  water  flowing  through 
the  channel  per  second,  we  have 


Q  =  av  =  92.26a^^J^'  (6) 


Cor, — For  pipes,  we  have 


Ip  Ind  _  4:1 

"a  ~  f^~  d' 


326  COEFFICIENTS  OF  FRICTION, 

which  in  (3)  ffives 

A  =  0.030261^,  (7) 

which  agrees  with  (5)  of  Art.  104  and  (1)  of  Art  105. 


EXAMPLE. 

How  much  fall  must  a  canal,  whose  length  is  2600  ieet, 
whose  lower  width  is  3  feet,  whose  upper  width  is  7  feet, 
and  whose  depth  is  3  feet,  have  in  order  to  carry  40  cubic 
feet  of  water  per  second  ?    Here  we  have 


^  =  3  4-  2V2M-  32  =  10.211, 


and 


40 
15 


Substituting  in  (5),  we  have 


h  =  0.00011747 


2600  X  10.211  /8\2 


15 

0.305422  X  10.211  x  64 
15x9 


(ir 


=  1.48  feet. 


122.  Coefficients  of  Friction. — The  coeflficient  of 
friction  /varies  greatly  with  the  degree  of  roughness  of  the 
channel  sides,  and  somewhat  also  with  the  velocity,  as  in 
the  case  of  pipes,  increasing  slightly  when  the  velocity 
diminishes,  and  decreasing  when  the  velocity  increases.  A 
common  mean  value  assumed  for  /  is  0.007565,  which  we 
used  in  the  last  Art.,  though  it  has  quite  a  range  of  values. 
Weisbach,  from  255  experiments,  obtained  for  /  the  follow- 
ing values  at  different  velocities : 


EXAMPLE. 


227 


V   = 

0.3 
0.01215 

0.4 
0.01097 

0.5 
0.01035 

0.6 
0.00978 

0.7 
0.00944 

V   = 

f  = 

0.8 
0.00918 

0.9 
0.00899 

1.0 
0.00883 

0.00836 

2 
0.00812 

V   = 

3 

0.00788 

5 
0.00769 

7 
0.00761 

10 
0.00755 

15 

0.00750 

In  using  this  table  for  the  value  of/  when  v  is  not  known, 
we  must  proceed  by  approximation.  Determine  v  approxi- 
mately from  (4)  of  Art.  121.  Then  from  this  value  of  v 
find /by  means  of  the  table,  and  substitute  the  value  of  / 
so  found  in  (2),  and  determine  a  new  value  of  v. 


EXAMPLE. 

What  must  be  the  fall  of  a  canal  1500  feet  long,  whose 
lower  width  is  2  feet,  upper  width  8  feet,  and  depth  4  feet, 
when  it  is  required  to  convey  70  cubic  feet  of  water  per 
second  ? 

Here  we  have 

p  =  2  +  2VI6  +  9  =  13, 
«  =  5  X  4  =  20, 

v  =  ^  =  3.5; 

hence,  from  the  table, 

/  =  0.00784. 

Substituting  in  (1)  of  Art.  121,  we  have 

A  =  0.00784  i£20^><S? 
20  2g 


86.436 
^6474" 


=  1.34  feet. 


228 


VARIABLE  MOTION. 


123.  Tariable  Motion. — In  every  stream  in  which  the 
discharge  is  constant  for  a  given  time,  the  velocity  at  differ- 
ent places  depends  on  the  slope  of  the  bed.  In  general,  the 
velocity  will  he  greater  as  the  slope  of  the  bed  is  greater; 
and,  as  the  velocity  varies  inversely  as  the  transverse  section 
of  the  stream,  the  section  will  be  least  where  the  velocity 
and  slope  are  greatest.  In  a  stream  in  which  the  velocity 
is  yariable,  the  work  due  to  the  fall  of  the  stream  for  a  given 
distance  is  equal  to  the  work  destroyed  by  friction  together 
with  the  kinetic  energy  corresponding  to  the  change  of 
velocity,  i.  c,  the  whole  fall  is  the  sum  of  that  expended  in 
overcoming  friction,  and  of  that  expended  in  increasing  the 
velocity,  when  the  velocity  increases,  or  if  the  velocity  de- 
creases, the  head  is  the  difference  of  these  quantities.* 

The  resistance  of  friction  upon  a  small  portion  of  the 
length  of  the  stream  may  be  regarded  as  constant  and  meas- 
ured by  a  head  of  water 


~^  a2g' 


(1) 


Fig.  64 


Let  ABCD  represent  a  longitudi- 
nal section  of  a  short  portion  of  a 
stream,  AB  the  surface  of  the  stream, 
and  AE  and  HG  two  horizontal 
lines.  Let  I  =  the  length  of  AB  in 
feet;  h  =  BE,  the  fall  from  A  to 
B  ;  i^o  =  the  velocity  of  the  stream  at  the  upper  section 
AD  ;  and  v^  =  the  velocity  at  the  lower  section  BC. 

Now  the  velocity  of  any  particle  B,  at  the  surface  of  the 
stream,  is  due  to  the  height  ?i,  together  with  the  velocity  at 
A ;  hence  we  have,  for  its  velocity  v^, 


^  —  A  4-  ^. 


(2) 


*  In  long  rivers,  with  elopes  not  greater  than  3  feet  per  mile,  the  velocity  head 
is  usually  insignificant  compared  with  the  friction  head.  (See  Fanning's  Water- 
Supply  Engineering,  p.  303.; 


VARIABLE  MOTION.  229 

Any  particle  G,  beneath  the  surface  ol  the  water,  is 
pressed  forward  by  the  head  AH  =  EG,  and  pressed  back- 
ward by  the  head  BG;  hence  the  head  which  produces  mo- 
tion is  EG  —  BG  =  EB,  or  h,  as  before ;  and  therefore  (2) 
is  true  for  any  particle.  Solving  (2)  for  h,  and  adding  the 
resistance  of  friction,  as  given  in  (1),  we  have 

n  =  ^-l^  +  fk^,  (3) 

in  which  p,  a,  and  v  denote  the  mean  values  of  the  wetted 
perimeter,  the  transverse  section,  and  the  velocity,  respect- 
ively. 

If  Uq  and  «i  denote  the  areas  of  the  upper  and  lower 
transverse  sections,  respectively,  and  Q  the  quantity  of 
water  which  flows  through  any  section  in  a  unit  of  time,  we 
have 

a  — ,  (4) 

and  Q  =  af,v^  =  a^v^.  (6) 

From  (5),  we  have 

v,^  —  V, 


'^g 


o' 


Vai2      a^V  2g'  ^"' 


Now  if  the  water  flowed  with  the  velocity  v„,  we  would 
have  the  head  due  to  the  resistance  of  friction,  from  (1), 

and  if  it  flowed  with  the  velocity  v^,  we  would  have  the 
head  due  to  the  resistance  of  friction 

But  the  former  expression  is  less,  and  the  latter  is  greater 
than  the  true  head  due  to  the  resistance  of  friction ;  henqe, 


230     ,  VARIABLE  MOTION. 

the  mean  of  these  results  will  give  the  friction  head  approx- 
imately. Therefore,  taking  the  mean  of  (7)  and  (8)5  and 
substituting  for  a  its  value  from  (4),  we  have,  for  the  re- 
sistance of  friction, 

[from  (5)]. 

Substituting  (6)  and  (9)  in  (3),  we  have,  for  the  whole 
head, 


7i  = 


"---+/--^(-  +  -)1^'.   (10) 


Solving  (10)  for  Q,  we  have 

Q V2p 

/irrXTTZSTTrTTTV   ^^^^ 

In  a  prismoidal  channel  it  will  be  a  sufficiently  close  ap- 
proximation to  the  truth  to  assume  that  the  surface  line  of 
the  water  is  straight,  and  then  from  this  assumption  to  com- 
pute the  transverse  sections  and  their  perimeters.  When 
we  have  these,  with  the  quantity  of  water  carried  and  the 
length  of  a  portion  of  the  river  or  canal,  we  may  determine 
the  corresponding  fall  li  by  (10)  ;  and  when  we  have  the 
length,  fall,  and  cross-section,  we  may  determine  the  quan- 
tity Q  by  means  of  (11).  Where  greater  accuracy  is  re- 
quired, we  should  calculate  li  or  Q  for  several  small  portions 
of  the  stream,  and  then  take  the  anthmetic  mean  of  the 
results.    If  only  the  total  fall  is  known,  this  value  should 

be  substituted  for  h  in  (11),  and  instead  of  — ^ g  we 


EXAMPLE.  231 

should  use  — ^ 5,  where  an  denotes  the  area  of  the  low- 

est  transverse  section,  and  instead  of 

the  sum  of  all  the  similar  values  for  the  different  portions 
of  the  stream  should  be  used.  (See  "Weisbach's  Mechs., 
p.  969 ;  also  Tate's  Mech.  Phil,  p.  305.) 

BX  AM  PLE. 

A  stream  falls  9.6  inches  in  300  feet,  the  mean  value  of 
its  wetted  perimeter  is  40  feet,  the  area  of  its  upper  trans- 
verse section  is  70  square  feet,  and  that  of  its  lower  is  60 
square  feet.     Find  the  discharge  of  this  stream. 

From  (11),  we  have 

^  _  8.025\/a8 

V  — 


/  1  1        ^...^^^..r  300x40/  1         1  \ 

=  354|  cubic  feet. 


a/0.0000737  +  0.0003365 
The  mean  velocity  is 

=  :r^^  =  5.45  feet; 


a^  +  a^        130 

hence  (Art.  122),  a  more  accurate  value  of /is  0.00768,  and 
therefore  we  have 

Q  =r ^'-^^ =  352.5. 

VO.  0000731  +  0.0003416 

If  the  same  stream  has  a-t  another  place  a  fall  of  11  inches 
in  450  feet,  and  if  the  area  of  its  upper  transverse  section  is 


232   BOTTOM    VELOCITY  AT    WHICH  SCOUR    COMMENCES. 

50,  and  that  of  its  lower  60  square  feet,  the  mean  value  of 
its  wetted  perimeter  being  36  feet,  we  have 

„ 8.025  \/a9r67      - 

=  8  02-     /  Q'91^'^ 

'     ^V  —  0.0001222  +  0.0007549 

=  305^  cubic  feet. 

The  mean  of  these  values  is 

ScH. — The  following  is  Chezy's  formula,  with  three  dif- 
ferent coefficients,  varying  from  69  "for  small  streams 
under  2000  cubic  feet  per  minute,"  to  96  "for  large  rivers 
such  as  the  Clyde  or  the  Tay." 

y  =  69  (r  sin  d)k         For  small  streams. 

V  =  93  (r  sin  6)^.        Eytelwein's  coefficient. 

V  =z  96  (r  sin  6)'^.        For  large  streams. 

124.  Bottom  Telocity  at  which  Scour  Com- 
mences.— A  river  channel  is  said  to  have  a  fixed  regimen, 
when  it  changes  little  in  draft  or  form  in  a  series  of  years. 
In  some  rivers,  the  deepest  part  of  the  channel  changes  its 
position  perpetually,  and  is  seldom  found  in  the  same  place 
two  successive  years.  The  sinuousness  of  the  river  also 
changes  by  the  erosion  of  the  banks,  so  that  in  time  the 
position  of  the  river  is  completely  altered.  In  other  rivers, 
the  change  from  year  to  year  is  very  small,  but  probably  the 
regimen  is  never  perfectly  fixed  except  where  the  rivers  flow 
over  a  rocky  bed.     If  a  river  had  a  constant  discharge,  it 


TRANSPORTING   POWER    OF   WATER.  233 

would  gradually  modify  its  bed  till  a  permanent  regimen 
was  established.  But  as  the  volume  discharged  is  constant- 
ly changing,  and  therefore  the  velocity,  silt  is  deposited 
when  the  velocity  decreases,  and  scour  goes  on  when  the 
velocity  increases  in  the  same  place. 

It  has  been  found  by  experiment  *  that  a  stream  moving 
with  a  velocity  of  3  inches  per  second  will  carry  alongj^we 
clay  and  soft  earth  ;  moving  6  inches  per  second,  will  carry 
loam  ;  1  foot  per  second,  will  carry  sand;  2  feet  per  second, 
gravel  j  3^  feet,  pebbles  an  inch  in  diameter  ;  4  feet,  broken 
stone, flint ;  5  feet,  chalk,  soft  shale;  6  feet,  rock  in  beds; 
10  feet,  hard  rock. 

125.  Transporting  Power  of  Water.— The  specific 
gravity  of  rocks  varies  from  2.25  to  2.64;  when  immersed 
in  water,  therefore,  they  lose  nearly  half  their  weight.  This 
fact  greatly  increases  the  transporting  power  of  water.  The 
pressure  of  a  current  of  water  against  any  surface  varies  as 
the  square  of  the  velocity  and  as  the  area  of  the  surface  f 
(Art.  97),  But  in  similar  figures,  surfaces  vary  as  the 
squares  of  the  diameters ;  hence,  the  pressure  of  the  current 
varies  as  the  square  of  the  velocity  and  as  the  square  of  the 
diameter,  i.  e.,  the  pressure  of  the  current  against  a  surface 
varies  as  the  square  of  its  velocity  multiplied  by  the  square 
of  the  diameter  of  the  surface.  Calling  P  the  pressure 
which  the  current  exerts  against  a  rock,  v  its  velocity,  and 
d  the  diameter  of  the  surface  of  the  rock,  we  have 

P  a  t^xcP:  (1) 

Now  the  resistance  to  be  overcome,  or  the  loeight  of  the 
rock,  varies  as  the  cube  of  the  diameter;  i.  e.,  calling  W 
the  weight  of  the  rock,  we  have 

TT'  OL  fZ3.  (2) 

*  Experiments  byDnbnat.    Sec  Ency.  Brit,  Vol.  XII.,  p.  503. 
+  Supposing  that  the  area  of  the  cross-section  of  the  stream  is  at  least  large 
enough  to  cover  the  surface. 


234 


TRANSPORTING   POWER   OF  WATER. 


But  when  the  current  is  just  able  to  move  the  rock,  we 
have 

P  o:   W.  (3) 

Therefore,  from  (1),  (2),  and  (3), 

d^  <x  v^xcP ; 

.:     d  Qc  v^, 

which  in  (1)  gives  P  cc  v^  x  v^, 

or  P  oc  v\  (4) 

That  is,  the  transporting  power  of  a  current  varies 
as  the  sixth  power  of  the  velocity. 

This  may  also  be  shown  geometrically  as  follows  :  Let  a 
represent  a  cubic  inch  of  stone,  which  a  current  of  given 
velocity  will  just  move,  and  let  J  be  a  cube 
of  stone  64  times  as  large.     Now  if  the 
velocity   of  the   current   be   doubled,  the 
force  against  each  square  inch  of  h  will  be 
four  times  as  great  as  that  against  a ;  but 
the  surface  of  h  opposed  to  the  current  is 
sixteen  times  as  great  as  that  of  a,  and  the 
pressure  would  be  increased  sixteen  times 
from  this  cause  ;  therefore  the  whole  press- 
ure against  h  from  these  two  causes  would 
be  4  X 16  =  64  times  as  great  as  against  a. 
But  the  weight  also  of  h  is  (54  times  as  great  as  that  of  a; 
therefore  the  current  would  be  just  able  to  move  it. 

We  have  seen  (Art.  124)  that  a  current  3|-  feet  per  second, 
or  about  two  miles  an  hour,  will  move  pebbles  an  inch  in 
diameter,  or  about  three  ounces  in  weight.  It  follows  from 
the  above  law  that  a  current  of  ten  miles  an  hour  will  bear 
fragments  of  1\  tons,  and  a  torrent  of  20  miles  an  hour  will 
carry  fragments  of  100  tons  in  weight.* 


\\ 


^s^ 


Fig.  65 


*  Le  Conte's  Geology,  p.  18. 


BACK    WATER. 


235 


126.  Back  Water. — When  a  dam  is  built  across  a  stream 
80  as  to  raise  the  water  and  form  a  pond,  the  surface  of  the 
water  in  the  pond  will  not  be  horizontal.     Let  AB  represent 


a  dam,  and  C  the  surface  of  the  water  directly  over  the  dam. 
If  the  horizontal  line  CD  be  drawn  from  the  surface  at  C 
to  the  point  D,  where  it  intersects  the  natural  surface  of 
the  stream,  the  surface  of  the  water  in  the  pond  will  be 
everywhere  above  this  line,  except  at  0,  its  height  increasing 
as  the  distance  from  the  dam  increases,  and  this  elevation 
may  extend  for  quite  a  distance  up  the  stream  above  the 
point  D. 

The  elevation  CDFE  above  the  horizontal  CD  is  called 
back  water.  As  the  stream  approaches  the  horizontal  sur- 
face DC,  its  velocity  is  diminished,  because  the  slope  on 
which  the  velocity  depends  is  very  small,  and  as  the  velocity 
is  diminished  the  water  is  heaped  up  above  DC,  even  ex- 
tending up  the  stream,  until  the  slope  is  sufficient  for  the 
water  to  flow  off.  When  this  slope  is  established,  the  stream 
FEC  flows  smoothly  along  its  liquid  channel. 

Fig.  QQ  shows  a  longitudinal  section  of  the  river  Weser, 
in  Germany,  where  a  dam  was  built.  The  mean  depth  of 
the  stream  was  about  2.5  feet,  the  surface  was  raised  7.5 
feet,  the  slope  of  the  stream  was  quite  uniform  for  a  distance 


236  RIVER  BENDS. 

.  of  ten  miles.  At  the  point  D,  three  miles  from  the  dam,  it 
was  found  by  measurement  that  the  surface  E  was  elevated 
over  15  inches  above  D.  At  a  distance  of  4  miles  above  the 
dam,  the  surface  was  elevated  by  the  dam  9  inches.* 

127.  River  Bends.— When  rivers  flow  in  narrow  val- 
leys, where  the  banks  do  not  readily  yield  to  the  action  of 
the  current,  the  effect  of  any  variation  of  velocity  is  only 
temporarily  to  deepen  the  bed.  In  wide  valleys  and  allu- 
vial plains,  where  the  soil  of  the  banks  is  more  easily  Avom 
by  the  current  than  the  bottom,  any  increase  in  the  volume 
of  the  water  will  widen  the  bed ;  and  if  one  bank  yields 
more  than  the  other,  icmdings  or  hends  will  be  formed,  and 
these  windings  which  are  thus  formed  tend  to  increase  in 
curvature  by  the  scouring  away  of  material  from  the  outer 
bank  and  the  deposition  of  detritus  along  the  inner  bank. 
The  windings  sometimes  increase  till  a  loop  is  formed,  with 
only  a  narrow  strip  of  land  between  the  two  encroaching 
branches  of  the  river.  Finally,  a  "  cut-off"  may  occur,  a 
waterway  being  opened  through  the  strip  of  land,  and  the 
loop  left  separated  from  the  stream,  forming  a  lagoon  of 
marsh  shaped  like  a  horse-shoe. 

It  is  usually  supposed  that  the  water,  tending  to  go  for- 
wards in  a  straight  line,  rushes  against  the  outer  bank  and 
scours  it,  at  the  same  time  creating  deposits  at  the  inner 
bank.  This  view  is  considered  by  many  engineers  as  very 
incomplete.  Prof.  James  Thomson  has  given  an  explana- 
tion of  the  action  at  a  bend,  which  he  has  completely  con- 
firmed by  experiment,  f  He  thinks  that  the  scouring  at  the 
outer  side  and  the  deposit  at  the  inner  side  of  the  bend  are 
due  to  the  centrifugal  force,  in  virtue  of  which  the  water 
passing  round  the  bend  presses  outwards,  and  the  free  sur- 
face in  a  radial  cross-section  has  a  slope  from  the  inner  side 
upwards  to  the  outer  side. 

♦  D'Aubuisson'g  Hydraulics,  Art.  166. 

t  Proc  Inst,  of  Mech.  Engineers,  1879,  p.  456. 


EXAMPLES,  237 

EXAMPLES. 

1.  A  thin  plane  area  moves  edgeways  through  the  water, 
in  which  it  is  completely  immersed.  Find  the  resistance 
per  square  foot  at  a  speed  of  20  miles  per  hour. 

Alls.  3.442  lbs. 

2.  The  length  of  a  pipe  is  400  feet,  the  head  of  water  is 
6  feet,  and  the  diameter  of  the  pipe  is  6  inches,  the  entrance 
to  it  being  cyHndrical.  Find  (1)  the  head  due  to  friction, 
(2)  the  velocity  of  discharge,  and  (3)  the  quantity  discharged 
per  second. 

Take  /  =  0.03,  g  =  32,  and  use  (3)  of  Art,  104  for  v. 

Ans.   (1)  5.646  ft. ;  (2)  3.88  ft. ;  (3)  0.7619  cu.  ft. 

3.  When  the  pipe  is  800  feet  long,  the  head  of  water  12 
feet,  and  the  diameter  6  inches,  find  (1)  the  friction  head, 
(2)  the  velocity  of  discharge,  and  (3)  the  quantity  dis- 
charged per  second. 

Ans.   (1)  11.635  ft.  ;  (2)  3.039  ft.;  (3)  0.7734  cu.  ft. 

4.  When  the  pipe  is  1600  feet  long,  the  head  24  feet,  and 
the  diameter  6  inches,  find  the  same  quantities  as  in  the  last 
two  examples. 

Ans.  (1)  23.63  ft.;  (2)  3.969  ft.  ;  (3)  0.7793  cu.  ft. 

5.  When  the  pipe  is  3200  feet  long,  the  head  48  feet,  and 
the  diameter  6  inches,  find  the  same  quantities. 

Ans.   (1)  47.627  ft.  ;  (2)  3.984  ft. ;  (3)  0.7823  cu.  ft.* 

6.  When  the  pipe  is  800  feet  long,  the  head  12  feet,  and 
the  diameter  5  inches,  find  the  same  three  quantities  as 
before. 

Ans.  (1)  11.694  ft.;  (2)  3.605  ft;  (3)  0.4915  cu.  ft. 


*  An  inspection  of  Exs.  2.  3,  4,  and  5,  shows  that  if  a  6-inch  pipe  be  laid  with  a 
uniform  slope  of  6  feet  in  400  feet,  nearly  all  the  head  is  consumed  by  friction,  so 
tliat  only  a  very  small  fraction  of  the  entire  head  remains  to  generate  the  final  veloc- 
ity and  to  overcome  the  resistance  at  the  entrance  to  the  pipe,  i.e.,  in  each  case 
there  is  only  about  0.35  of  a  foot  of  head  left ;  one-third  of  this  is  expended  in  over- 
coming the  resistance  at  the  entrance  to  the  pipe,  and  the  other  two-thirds  in  pro- 
ducing  velocity. 


238'  EXAMPLES. 

7.  When  the  leugth  is  1600  feet,  the  head  24  feet,  and 
the  diameter  5  inches,  find  the  same  quantities. 

Ans.   (1)  23.69  ft.;  (2)  3.628  ft.  ;  (3)  0.4947  cu.  ft. 

8.  When  the  length  is  800  feet,  the  head  5  feet,  and  the 
diameter  6  inches,  find  the  same  quantities. 

Ans<.   (1)  4.848  ;  (2)  2.542;  (3)  0.499. 

9.  When  the  length  is  800  feet,  the  head  16  feet,  and  the 
diameter  6  inches,  find  the  same  quantities. 

Ans.   (1)15.514;  (2)4.548;  (3)0.893. 

10.  Two  pipes  of  the  same  length  are  3  inches  and 
4  inches  in  diameter,  respectively.  Compare  the  losses  of 
head  by  friction,  (1)  when  the  velocity  is  the  same,  and  (2) 
when  they  deliver  the  same  quantities  of  water. 

Ans.  (1)  1.33  ;  (2)  4.21. 

11.  Water  is  to  be  raised  to  a  height  of  20  feet  by  a  pipe 
30  feet  long  and  6  inches  in  diameter.  What  is  the  greatest 
admissible  velocity  of  the  water,  if  not  more  than  10  per 
cent,  additional  power  is  to  be  required  in  consequence  of 
the  friction  of  the  pipe  ?  Ans.  ^^  feet  per  second. 

12.  Two  reservoirs  are  connected  by  a  pipe  6  inches  in 
diameter  and  f  of  a  mile  long.  For  the  first  quarter  mile 
the  pipe  slopes  at  1  in  50,  for  the  second  at  1  in  100,  while 
in  the  third  it  is  level.  The  head  of  water  over  the  inlet  is 
20  feet,  and  that  over  the  outlet  9  feet.  Neglecting  all  loss 
except  that  due  to  surface  friction,  find  (1)  the  velocity  per 
second,  and  (2)  the  discharge  in  gallons  per  minute,  assum- 
ing /  =  0.0348.  Ans.   (1)  3.43  ft;  (2)  253  gallons. 

13.  A  tank  of  250  gallons  is  50  feet  above  the  street.  It 
is  connected  with  the  street  main,  the  head  of  which  is  52 
feet,  by  a  pipe  100  feet  long.  (1)  Find  the  diameter  of  the 
pipe  that  the  tank  may  be  filled  in  20  minutes ;  (2)  what 
must  the  head  in  the  main  be  to  fill  the  tank  in  5  minutes 
with  the  jiipe  ?  Ans.   (1)  1.6  inches;  (2)  82  feet. 


EXAMPLES.  339 

14.  What  is  the  discharge  per  second  through  a  pipe  48 
feet  long  and  2  inches  in  diameter,  under  a  head  of  5  feet  ? 

Sue.— Assume  /  =  .02,  and  obtain  from  (8)  of  Art.  106,  v  =  6.6 
feet,  and  therefore  (Sch.  of  Art.  105),  /=  .0311,  which  in  (8)  of  Art. 
106  gives  V  =■  6.53  feet.     .  •.  etc. 

Ans.  245.8  cubic  inches. 

15.  What  must  be  the  diameter  of  a  pipe  100  feet  long, 
which  is  to  discharge  one-half  of  one  cubic  foot  of  water  per 
second  under  a  head  of  5  feet?  Aiis.  3.82  inches. 

See  remark  in  Art.  107. 

16.  If  the  diameter  of  one  portion  of  the  compound  pipe 
(Fig.  55)  is  twice  that  of  the  other,  and  if  the  velocity  of 
the  water  in  the  larger  is  10  feet,  find  (1)  the  coefficient  of 
resistance,  and  (2)  the  loss  of  head  at  the  sudden  enlarge- 
ment, the  water  flowing  from  the  small  pipe  into  the  large 
one.  Ans.  (1)  9 ;  (2)  13.95  feet. 

17.  A  pipe  2  inches  in  diameter  is  suddenly  enlarged  to 
3  inches.  If  it  discharge  100  gallons  per  minute,  the  water 
flowing  from  the  small  pipe  into  the  large  one,  find  (1)  the 
coefficient  of  resistance,  and  (2)  the  loss  of  head  at  the  sud- 
den enlargement.  Ans.   (1)  1.59;  (2)  8|^  inches. 

18.  In  the  last  example,  if  the  water  moves  in  the  reverse 
direction,  find  the  loss  of  head  caused  by  the  sudden  con- 
traction, assuming  the  coefficient  of  contraction  to  be  0.66. 

Ans.   7^  inches. 

19.  A  pipe  contains  a  diaphragm  with  an  orifice  in  it,  the 
area  of  which  is  one-fifth  the  sectional  area  of  the  pipe. 
Find  the  coefficient  of  resistance  of  the  diaphragm,  assum- 
ing the  contraction  on  passing  through  the  orifice  the  same 
as  that  at  efflux  from  a  vessel  through  a  small  orifice  in  a 
thin  plate.  Ans.  46.4. 

20.  A  horizontal  pipe  30  feet  long  is  suddenly  enlarged 
from  2  inches  to  3  inches,  and  then  suddenly  returns  to  its 
original  diameter  ;  the  length  of  each  section  is  10  feet.     If 


240  EXAMPLES. 

it  discharge  100  gallons  per  minute  into  the  atmosphere, 
find  the  total  loss  of  head,  assuming  the  coeflBcient  of  fric- 
tion /  =  .03.  Ans.  10  ft.  2|  ins. 

21.  Find  the  loss  of  head  in  inches  due  to  a  bend  in  a 
pipe  2  inches  in  diameter,  the  radius  of  curvature  being  G 
inches,  and  the  velocity  of  the  water  being  12  feet  per 
second.  Ans.  0.2  of  an  inch. 

22.  If  the  pipes  in  Ex.  2,  Art.  106,  which  are  to  discharge 
25  cubic  feet  of  water  per  minute,  contain  two  elbows,  each 
of  90°,  find  the  total  loss  of  head. 

Here  we  have,    h  =  (1.505  +  8.748  +  2x0. 984)  ^  =  etc. 

Ans.  1.76  feet. 

23.  If  the  pipe  in  Ex.  14  contains  5  bends,  the  radius  of 
curvature  of  each  being  2  inches,  find  (1)  the  velocity  of  the 
Avater  issuing  from  the  pipe,  and  (2)  the  quantity  discharged 
per  second. 

Here  /3  is  found  [from  (2)  of  Art.  Ill]  to  be  0.294.     .  • ,  etc, 

Ans.  (1)  5.964  ieet;  (2)  224.81  cubic  inches. 

24.  What  quantity  of  water  will  be  delivered  by  a  canal 
5800  feet  long,  when  the  fall  is  3  feet,  its  depth  5  feet,  its 
lower  breadth  4  feet,  and  its  upper  hreadth  12  feet  ? 

Here^  =  0.42015.     .'.  etc. 
a 

Atis.  129.48  cubic  feet. 

25.  Find  the  quantity  of  water  that  is  carried  by  a  stream 
40  feet  wide,  whose  mean  depth  is  4^  feet,  and  whose 
wetted  perimeter  is  46  feet,  when  it  falls  1  inch  in  75  feet. 

Here  v  approximately,  from  (4)  of  Art.  121,  =  6.1  feet.  .-.  f  = 
0.00765,  and  v  more  correctly  =  6.05  feet.     .  *.  etc. 

Ans.  1089  cubic  feet. 

26.  A  main  3100  feet  long  is  to  discharge  water  from  a 
reservoir  having  a  head  of  75  feet.  It  is  proposed  to  put  in 
a  6-inch  pipe  for  800  feet,  beginning  at  the  reservoir,  then  a 


EXAMPLES.  241 

5-inch  pipe  for  800  feet  more,  and  a  4:-inch  pipe  for  the 
remaining  1500  feet.  The  coefficient  of  friction  /  being 
0.034,  find  (1)  the  diameter  of  a  uniform  pipe  3100  feet 
long  having  the  same  friction,  and  (2)  the  velocity  of  dis- 
charge. 

Use  (7)  of  Art.  104  for  v. 

Ans.  (1)  4.427  inches ;  (2)  4874  feet. 

27.  If  in  the  last  example  the  first  pipe  of  the  main  is 
2000  feet  long  and  6  inches  in  diameter,  the  second  800 
feet  long  and  5  inches  in  diameter,  and  the  third  300  feet 
long  and  4  inches  in  diameter,  the  head  being  75  feet,  find 
(1)  the  diameter  of  the  equivalent  main,  and  (2)  the  velocity 
of  discharge.  Ans.  (1)  5.2  inches  ;  (2)  5.289  feet. 

28.  If  in  Ex.  26  the  pipe  is  6  inches  in  diameter  for  the 
whole  length  of  3100  feet,  the  head  being  75  feet,  find  the 
velocity  of  discharge.  Ans.  5.675  feet. 

29.  Into  a  branch  main  2000  feet  long,  6  inches  in  diam- 
eter, water  enters  with  a  velocity  of  15.27  feet  a  second;  1 
cubic  foot  of  water  is  delivered  into  service-pipes  for  every 
1000  feet  of  length,/  =  .0303,  what  is  the  loss  of  head  in 
the  2000  feet  ?  Ans.  211.53  feet. 


CHAPTER    III. 


MOTION     OF     ELASTIC     FLUIDS. 

128.  Work  of  the  Expansion  of  Air If  air  expands 

without  doing  any  work  its  temperature  remains  constant.* 
It  follows  from  this  that  as  air  changes  its  state,  the  inter- 
nal work  done  is  proportional  to  the  change  of  temperature. 
When,  in  expanding,  air  does  work  against  an  external 
resistance,  either  heat  must  be  supplied  or  the  temperature 
falls. 

Suppose  a  given  mass  of  air  to  be  confined  in  a  cylinder 
having  a  piston  of  one  square  foot  area.  Let  v^  be  the 
initial  volume  and  jo^  the  initial  pressure  of  the  air,  and 
suppose  the  piston  to  move  so  as  to  expand  the  air  to  any 
other  volume  v  with  pressure  p.  Then  if  heat  is  supplied 
to  the  air  during  the  expansion  so  that  the  temperature 
remains  constant,  we  have  (Art.  48), 

Vv^p^v^.  (1) 

Now  if  we  represent  the  pressures 
by  the  ordinates,  and  the  correspond- 
ing volumes  by  the  abscissas  of  a  curve 
AB  referred  to  the  axes  OX,  OY,  the 
curve  represents  the  relative  changes 
of  volume  and  pressure.     Then  OM  = 
Vj  and  MPj  =  jSj  is  a  point  Pj  corre- 
sponding to  a  volume  v^  and  pressure 
j9j.     Similarly  {v,p)  is  any  other  point  P  of  the  curve  cor- 
responding to  a  volume  v  and  pressure  jo;  and  since  each 
member  of   (1)   is  constant,   the  curve   is  a  rectangular 
hyperbola. 


*  This  reeolt  was  first  demonstrated  ezperimentally  by  Joule. 


WORK   OF  THE  EXPANSION   OF  AIR.  243 


The  work  of  expansion  between  the  pressures  jh  ''^^^  Pz 
is  represented  by  the  area  of  the  space  MPjPgN  (Anal. 
Mechs.,  Art.  232).  To  find  an  algebraic  expression  for  this 
work,  let  p  and  v  be  the  corresponding  pressure  and  volume 
at  any  intermediate  point  P  in  the  expansion.  Then  the 
work  done  on  the  piston  during  the  expansion  from  v  to 
v-\-dv  ispdv,  and  the  whole  work  done  during  the  expan- 
sion from  Vj  to  V2,  represented  by  the  area  MP^PgN, 

=   /pdv  =PiV,  I    '-J^[fi-om  (1)] 

=  Pi'^i  log  ~  =  Pi'^i  log*  ~r,  (2) 

V-i  Pz 

which  is  the  work  of  expanding  a  given  mass  of  air 
from  a  higher  pressure p ^  to  a  lower  pressure p^^. 

Cor. — In  order  to  compress  .  a  given  mass  of  air  whose 
volume  is  v^  and  whose  pressure  is  jOg,  into  a  volume  ^;^  of 
the  pressure  jjj,  the  work  to  be  done 

=  ^^2^8%^^.  (3) 

Pi 

which  is  the  work  of  compressing  a  given  mass  of 
air  from  a  lower  pressure  p„  to  a  higher  pressure  p^. 

ScH. — The  expressions  in  (2)  and  (3)  for  the  work  done 
during  the  expansion  and  compression  of  air,  are  correct 
only  when  the  temperature  of  the  air  remains  constant 
while  the  change  of  volume  or  density  is  taking  place ;  but 
the  temperature  of  the  air  remains  constant  only  when  the 
change  of  volume  takes  place  so  slowly  that  the  heat  in  the 
confined  air  has  sufficient  time  to  communicate  any  excess 
to  the  walls  of  the  vessel  and  to  the  exterior  air.  If  the 
change  of  density  occurs  so  quickly  that  it  is  accompanied 
by  a  change  of  temperature,  when  the  air  is  expanded  the 

•  Hyp.  log. 


244  VELOCITY  OF  EFFLUX   OF  ATR. 

temperature  is  lowered,  and  when  the  air  is  compressed  the 
temperature  is  increased.  Under  these  circumstances  the 
pressure  cannot  change  according  to  Art.  48,  and  other 
formulae  have  to  be  produced.  (See  Weisbach's  Mechs.,  p. 
936;  also  Encj.  Brit.,  Vol.  XII.,  p.  480.) 

129.  Velocity  of  Efflux  of  Air  According  to 
Mariotte's  Law. — Let  the  air  be  discharged  from  an 
orifice  with  the  velocity  v  feet  per  second ;  let  ?^  =  the 
weight  of  a  cubic  foot  of  air,  and  v^  ^  the  volume  of  air 
discharged  per  second;  then  the  work  performed  by  the 
volume  of  air  v^  in  passing  from  the  pressure  p^  to  the 
pressure  j'jg,  is,  by  (2)  of  Art.  128, 

i>i^i  log  ^, 

and  this  must  be  equal  to  the  work  stored  in  the  air  during 
the  efflux,  which  is 

Therefore,  we  have 


v/ 


.p^  l.„  Pi 


A  cubic  foot  of  air,  at  the  temperature  0°  of  the  centi- 
grade thermometer,  and  at  a  pressure  corresponding  to  the 
height  of  29.92  inches  of  the  barometer,  weighs  about 
0.08076  lbs.*  Therefore  for  any  temperature  t,  we  have 
for  the  weight  of  a  cubic  foot  of  air,  from  (2)  of  Art.  54, 
since  for  the  same  volume  and  temperature  the  weight 
varies  as  the  density, 

0.08076  ,^, 


♦  Determined  by  Regnault.    See  Weisbach's  Meclis.,  p.  795. 


VELOCITY  OF  EFFLUX   OF  AIR.  245 

If  the  pressure  differs  from  the  mean  pressure,  or  if  the 
height  of  the  barometer  is  not  29.93  inches,  but  h,  (2)  be- 
comes 

0.08076      b 


w  = 


1  +  at  29.92 
0.002699  h 


If  we  express  the  elastic  force  or  pressure  of  the  air  by  the 
pressure  ji;  upon  each  square  inch,  then  we  have 


which  in  (3)  gives 


29.92  ~  14.7' 


(3a) 


0.005494  » 

w  =  -— ~,  (4) 

1  +  at      '  ^  ^ 


p  _  (1  +  «0  144 
'■    w~      0.005494     ' 

where  p  is  the  pressure  on  each  square  foot. 
Substittjting  this  value  in  (1),  we  have 


(5) 


V  =  161.9  a/ 2^  (1  +  at)  log^. 


(6) 


If  b  is  the  height  of  the  barometer  and  h  that  of  the 
manometer  (Art.  46),  we  have 

^  -  ^-±^  (6a) 

which  in  (6)  gives 

V  =  1299  y^(l  +  at)  log  (^),  (7) 

where  v  is  the  velocity  in  feet,  b  the  hcigiit  of  the  barometer 
in  the  exterior  air,  h  the  height  of  the  manometer  for  the 
air  inside  the  vessel,  f  the  temperature  of  the  latter  in 
degrees  centigrade,  and  «  =  0.003665,  the  coeflBcient  of 
expansion  of  air  (Art.  53). 


246  VELOCITY  OF  EFFLUX   OF  AIR. 

CoE.  1. — If  the  pressures  p^  and  p^  are  nearly  equal  to 
each  other,  we  can  put 


-^m--(^-^i)- 


b  ^^2' 


which  in  (7)  gives 


i;  =  1299^(l+«0(l-4)t-  (8) 

When  T  is  very  small  (8)  becomes 

V  =  1299  a/(1  +  at)  -'  (9) 

CoE.  2. — Taking  g  =  32,  we  have  from  (1) 

V  =:8\/^\0g^'  (10) 

When  the  pressures  differ  but  little  from  each  other,  we 
may  obtain  approximate  formulae  as  follows  :  From  (10), 
we  have  

,  =  8^=lflog^.  (11) 

By  development  we  have 

logf^  =  log(l+^^^) 

=  ^.^^i!i_W/^^z:^i)  +  etc. 

Pi  ^     Pi      > 


Pi 


Neglecting  all  powers  of  — — —  above  the  first,  and  sub- 

Pi 
stituting  in  (11),  we  have 

>i  -Pi 


-  =  H/^-^^.  (12) 


EFFLUX   OF  MOVING   AIR,  247 


Neglecting  all  powers  of  — — ^-^  above  the  second,  we  have 


If  h  be  the  height  of  a  homogeneous  fluid,  of  the  same 
density  as  the  an-,  which  is  necessary  to  produce  the  pressure 
P\  —  jOgj  then^i  —  jOg  =  wh,  which  in  (12)  gives 

V  =  SVh.  (14) 

It  will  be  observed  that  (8),  (9),  (12),  (13),  (14)  are  true 
only  when  the  pressures  j9i  and  jSg  ^^'^  nearly  equal  to  each 
other. 

130.  Efflux  of  Moving  Air.— To  find  the  velocity  of 
efflux  ivheii  the  pressure  of  the  air  is  given  in  the 
pipe  through  which  it  flows. 

The  formulfB  for  efflux  found  in  Art.  129,  are  based  upon 
the  supposition  that  tlie  pressure  jo,  or  the  height  h,  of  the 
manometer  is  measured  at  a  place  where  the  air  is  at  rest, 
or  moving  very  slowly.  If  the  pressure  be  measured  at  a 
point  where  the  air  is  in  motion,  in  determining  the 
velocity  of  efflux,  we  must  take  into  account  the  kinetic 
energy  of  the  moving  air. 

Let  jOj  be  the  pressure  of  the  air  in  the  pipe  A,  as  indi- 
cated by  the  manometer  M,  and 
t\  the  velocity  of  the  air  passing     _____^^ 
the  orifice  of  the  manometer;  2^^ 

the  pressure  of  the  air  at  efflux,  - .         ^ 

and  ^2  its  velocity;  a^  the  area 


of  the  section  of  the  pipe  A,  and  p.    gg 

^2  the  area  of  the  orifice  0 ;  w  the 

weight  of  a  cubic  foot  of  the  air,  and  Q  the  volume  dis- 
charged per  second. 

Then  the  work  stored  in  the  air  while  passing  from  the 
pipe  A  to  the  orifice  0 


248  EFFLUX   OF  MOVING    AIR. 

and  this  must  equal  the  work  doue  by  the  expansion  of  the 
air  from  p^  io  p^.  Therefore,  from  (2)  of  Art.  128,  we 
have 

^(V-V)=^i^log^-  (1) 

The  volume  of  air  passing  through  the  pipe  per  second  is 
a-^v^,  and  that  which  passes  the  orifice  is  a^v^ ;  hence  we 
have  (Art  48) 

which  in  (1)  and  reducing,  gives 


/     '^gpx  log  ^ 


V  '"t-(^f-:)1 


which  is  the  velocity  of  efflux. 

Cor. — Substituting  for  the  numerator  its  value  as  given 
in  (8)  of  Art  129,  we  have 


or  approximately,  when  jOj  is  not  much  greater  than  jOgj 


V,  =  1399      /    tA-  (4) 


i^J 


COEFFICIENT  OF  EFFLUX.  349 

131.  Coefflcieiit  of  Efflux.— When  air  issues  from  an 
orifice,  the  section  of  the  current  undergoes  a  contraction 
similar  to  that  observed  in  the  efflux  of  water  (Art.  91).  If 
the  orifice  of  efflux  is  in  a  thin  plate,  the  stream  of  air  has 
a  smaller  cross-section  than  the  orifice,  and  the  practical 
discharge  is  less  than  the  theoretical. 

Denoting  the  coefficient  of  contraction  by  «,  we  have,  as 
in  the  case  of  water  (Art.  92),  «  =  the  ratio  of  the  cross- 
section  of  the  stream  of  air  to  that  of  the  orifice. 

Denoting  the  coefficient  of  velocity  by  0,  Ave  have,  as  in 

v 
Art.  93,  0  1=—,  where  v.  is  the  actual  and  v  the  theoret- 
V  * 

ical  velocity  of  discharge. 

Denoting  the  coefficient  of  efflux  by  jw,  we  have^  as  in  Art. 

94,  /i  =  ^  ^  «0,  where    Q^  is  the  actual  and    Q^  the 

theoretical  discharge. 

The  older  experiments  upon  the  efflux  of  air  through  ori- 
fices vary  considerably  from  each  other.  According  to  the 
experiments  of  Koch,*  /it  =  0.58  when  the  air  issues  from  an 
orifice  in  a  thin  plate  ;  /*  =  0.74  when  the  air  issues  from  a 
pipe  about  six  times  as  long  as  it  is  wide;  and  \i  =  0.85 
when  the  air  issues  from  the  conical  nozzle  of  a  bellows 
about  five  times  as  long  as  it  is  wide  and  having  a  lateral 
convergence  of  6°.  D'Aubuisson,  Poncelet,  and  Pecqueur 
found  values  somewhat  difEerent.f 

Weisbach  has  found  the  following  values  of  the  coefficient 
of  efflux  /* :  I 

Conoidal  mouth-pieces  of  the  form  of  the 

contracted    vein,    with    effective  fi  = 

pressures  of  0.33  to  1.1  atmosphere,         0.97    to  0.99 

*  Tate's  Mech.  Phil.,  p.  329.  t  Weisbach's  Mechs.,  p.  945. 

t  Ency.  Brit.,  Vol.  XII.,  p.  481. 


250  COEFFICIENT  OF  EFFLUX. 

Circular,  sharp-edged  orifices,   ....  0.563  to  0.788 

Short,  cylindrical  mouth-pieces,     ...  0.75 
Conical  pipes,  whose  angle  of  convergence 

is  about  6°, 0.92     ^ 

Conical   converging   mouth-pieces,  well 

rounded  off,   ........  0.98 

132.  The  Quantity  Discharged.— Let  h  be  the  area 
of  the  orifice  in  square  feet,  and  Q^  the  discharge  per  sec- 
ond in  cubic  feet  at  the  pressure  of  the  external  air.  Then 
we  have,  from  (1)  of  Art.  129, 


e,=,^y^2,|i-iog|i 


=  1299^/5:^/0"+  at)  log  (^-^)  (1) 

[from  (7)  of  Art.  129]  ; 
or,  from  (8)  and  (9)  of  Art,  129,  we  have 


_         Q,  =  1299iuA;y^(l  +  at)  (l  -  ^)  |  (2) 

=  1299//X;y/(i  +  «^)  I  (3) 

When  Q^  is  reduced  to  the  pressure  of  the  air  inside  the 
vessel,  we  have,  from  Art.  48, 


.-.     Q,  =  1299,./fe^y'(l  +  at)  log  (*-ti).  (4) 
From  (4)  of  Art.  130,  we  have 


Q  =  1299/.^      / ^.  (5) 


COEFFICIENT  OF  FRICTION  OF  AIR.  251 

133.  Coefficient  of  Friction  of  Air.— When  air  flows 
through  a  long  pipe,  it  has,  like  water,  a  resistance  of  fric- 
tion to  overcome,  due  to  the  surface  of  the  pipe ;  and  this 
resistance,  which  is  found  to  consume  by  far  the  greater 
part  of  the  work  expended,  can  be  measured  by  the  height 
of  a  column  of  air,  which  is  determined  by  the  expression, 

7  «2 

in  which,  as  in  the  case  of  water  (Art.  103),  I  denotes  the 
length,  d  the  diameter  of  the  pipe,  v  the  velocity  of  the  air, 
and  /  the  coefficient  of  resistance  of  friction,  to  be  deter- 
mined by  experiment.  The  work  expended  in  friction  gen- 
erates heat,  the  most  of  which  must  be  developed  in  the  air 
and  given  back  to  it.  Some  heat  may  be  transmitted 
through  the  sides  of  the  pipe  to  surrounding  materials,  but, 
in  all  the  experiments  that  have  thus  far  been  made,  the 
amount  so  conducted  away  appears  to  be  very  small ;  and  if 
no  heat  is  transmitted,  the  air  in  the  pipe  must  remain  sen- 
sibly at  the  same  temperature  during  expansion ;  that  is, 
the  heat  generated  by  friction  exactly  neutralizes  the  cool- 
ing due  to  the  work  done. 

A  discussion  by  Prof.  Unwin  *'  of  the  experiments  by  Messrs.  Culley 
and  Sabine  on  the  rate  of  transmission  of  light  carriers  through  pneu- 
matic tubes,  in  which  there  is  a  steady  flow  of  air  not  sensibly  affected 
by  any  resistances  other  than  the  surface  friction,  furnished  the  value 
/  =  0.028.  The  pipes  were  of  lead,  slightly  moist,  2\  inches  (0.187 
ft.)  in  diameter,  and  in  lengths  of  2000  to  nearly  6000  feet. 

Girard's  experiments  upon  the  motion  of  air  in  pipes  gave  a  mean 
coefficient  of  resistance,  /  =  0.0256 ;  those  of  D'Aubuisson  gave  as  a 
mean,  /  =  0.0238  ;  while  those  of  BufF  gave  the  mean  value  of  /  = 
0.0375. 

According  to  the  experiments  of  "Weisbach,  it  is  only  when  veloci- 
ties are  about  80  feet  that  the  coefficient  of  resistance  can  be  put  — 
0.034,  and  it  diminishes  as  the  velocity  of  the  air  in  the  pipe  increases. 

♦  See  Enqy.  Brit.,  Vol.  XJI.,  p.  49J. 


252  310TI0N  OF  AIR   IN  LONG    PIPES. 

He  found  that  the  coeflBcient  of  friction,  when  the  velocity  was  given 
in  feet,  could  be  expressed  approximately  by  the  following  formula. 

The  resistance  caused  by  elbows  and  bends  is  to  be  treated  in  the 
same  way  as  in  the  case  of  water  (Arts.  110,  111). 

134.  Motion  of  Air  in  Long  Pipes. — By  the  aid  of 

tlie  coefficient  of  friction  of  a  pipe,  we  can  calculate  the 
velocity  of  efflux  and  the  discharge  for  a  given  length  and 
diameter  of  the  pipe. 

Let    V  =  the  velocity  of  discharge  =  x' 

Vj  =  the  velocity  of  the  air  in  the  pipe. 
d  =  the  diameter  of  the  orifice,  whose  area  there- 
fore is  ^  =  ^T^d^. 
d^  =  the  diameter  of  the  pipe. 
|3q  =  the  coefficient  of  resistance  at  the  entrance  to 

the  pipe. 
/  =  the  coefficient  of  resistance  due  to  the  friction 

of  the  pipe. 
jSi  =  the  coefficient  of  resistance  at  the  orifice, 
jt?!  =  the  pressure  of  the  air  when  it  is  discharged. 
w  =  the  weight  of  a  cubic  foot  of  air. 
h  =  the  height  of  the  manometer  in  the  reservoir. 
b  =  the  height  of  the  barometer. 
I  =  the  length  of  the  pipe. 

Then  the  height  due  to  the  resistance  at  the  entrance  to 
the  pipe, 

- R ^- R ^^ 
-Po   2^    -   ^0  ^^4  2g' 

The  height  due  to  the  resistance  of  friction  in  the  pipe 
~'cli  2g   ~-'  d^  di*2g' 


MOTION  OF  AIR   IN  LONG   PIPES,  253 

The  height  due  to  the  resistance  at  the  orifice 
The  height  due  to  the  velocity 
Therefore,  the  total  height 

Also,  the  total  lieight,  from  (1)  and  (6a)  of  Art.  139, 

=  ^  ^^s{^  +  I)  =  ^-y  approximately.         (3) 

Therefore,  equating  (1)  and  (3),  and  solving  for  Q,  we 
have 


Q  =  k 


which,  from  (9)  of  Art.  139, 


=  1299—      /^ rVT^TTT— ^       (3) 


where  /3„  =  — „  -  1,  and  /3,  ==  — ^  -  1  (Art.  96). 

ScH. — In  Paris,  Berlin,  London,  and  other  cities,  it  has 
been  found  cheaper  to   transmit   messages   in   pneumatic 


254  LAW  OF  THE  EXPANSION  OF  STEAM. 

tubes  than  to  telegraph  by  electricity.  The  tubes  are  laid 
under  ground,  with  easy  curves ;  the  messages  are  made 
into  a  roll  and  placed  in  a  liglit  felt  carrier,  the  resistance  of 
which  in  the  tubes  in  London  is  only  |  oz.  A  current  of 
air,  forced  into  the  tube  or  drawn  through  it,  propels  the 
carrier.  In  most  systems  the  current  of  air  is  steady  and 
continuous,  and  the  carriers  are  introduced  or  removed 
without  materially  altering  the  flow  of  air. 

135.  The  Law  of  the  Expansion  of  Steam. — When 

steam  is  produced  in  a  close  vessel,  as  in  the  boiler  of  a 
steam  engine,  the  density  of  the  steam  increases  with  the 
temperature ;  but  so  long  as  the  temperature  remains  the 
same,  the  quantity  of  steam  that  can  be  raised  from  the 
water  is  limited,  and  the  steam  is  generated  at  its  maximum 
density  and  pressure  for  the  temperature,  whatever  this 
may  be;  if  the  temperature  falls,  a  portion  of  the  steam 
resumes  the  liquid  form,  and  the  density  of  the  steam  is 
diminished.  When  the  steam  is  in  its  condition  of  maxi- 
mum density,  it  is  said  to  be  saturated,  being  incapable  of 
vaporizing  or  absorbing  more  water  into  its  substance,  or 
increasing  its  pressure,  so  long  as  the  temperature  re- 
mains the  same.  Also,  on  the  contrary,  steam  will  not 
be  generated  with  less  than  the  maximum  quantity  of 
water  which  it  is  capable  of  appropriating  from  the  liquid 
out  of  which  it  ascends.  Any  change  in  either  one  of  the 
three  elements  of  pressure,  density,  or  temperature  of  steam 
is  necessarily  accompanied  by  a  change  of  the  other  two. 
The  same  density  is  invariably  accompanied  by  the  same 
pressure  and  temperature. 

If  the  volume  of  steam  over  water  be  increased,  while  the 
temperature  remains  constant,  then,  as  long  as  there  is  liquid 
in  excess  to  supply  fresh  vapor  to  occupy  the  increased 
space,  the  density  will  not  be  diminished,  but  will  remain 
constant  with  the  pressure.  If  the  source  of  heat  be  re- 
moved, when  all  the  liquid  is  evaporated,  the  pressure  and 


LAW  OF  TBE  EXPANSION  OF  STEAM.  255 

density  will  diminish,  when  the  volume  is  increased,  as  in 
permanent  gases;  and  if  the  volume  be  again  diminished, 
the  pressure  and  density  will  increase,  until  they  return  to 
the  maximum  due  to  the  temperature  ;  and  the  efiPect  of  any 
further  diminution  of  volume,  or  attempt  to  further  in- 
crease the  density  at  the  same  temperature,  is  simply 
accompanied  by  the  precipitation  of  a  portion  of  the  vapor 
to  the  liquid  state,  the  density  remaining  the  same. 

On  the  contrary,  if  the  application  of  heat  be  continued 
when  all  the  liquid  is  evaporated,  the  state  of  saturation 
ceases,  and  the  temperature  and  pressure  are  increased, 
while  the  density  remains  the  same  ;  the  steam  is  said  to  be 
superheated,  or  surcharged  with  heat,  and  it  becomes  more 
perfectly  gaseous.  While  in  this  condition,  if  it  were  to  be 
replaced  in  contact  with  water  of  the  original  temperature, 
it  would  evaporate  a  part  of  the  water,  transferring  to  it 
the  surcharge  of  heat,  and  would  resume  its  normal  state  of 
saturation. 

If  the  space  for  steam  over  the  water  remain  unaltered, 
then,  if  the  temperature  is  raised  by  the  addition  of  heat, 
the  density  of  the  vapor  is  increased  by  fresh  vaporization, 
and  the  elastic  force  is  consequently  increased  in  a  much 
more  rapid  ratio  than  it  would  be  in  a  permanent  gas  by  the 
same  change  of  temperature.  Conversely,  if  the  tempem- 
ture  be  lowered,  a  part  of  the  vapor  is  condensed,  the  den- 
sity is  diminished,  and  the  elastic  force  reduced  more 
rapidly  than  in  a  permanent  gas.  The  density  of  saturated 
steam  is  about  |  of  that  of  atmospheric  air,  when  they  are 
both  under  the  same  pressure  and  at  the  same  temperature. 

It  has  been  determined  experimentally  that  whatever  may 
be  the  pressure  at  which  steam  is  formed,  the  quantity  of 
fuel  necessary  to  evaporate  a  given  volume  of  water  is 
always  the  same  ;  also  the  relation  between  the  temperature 
and  pressure  of  saturated  steam  has  been  determined  experi- 
mentally, and  from  this  tables  have  been  formed  giving  the 
relation  between  the  pressure  and  volume  of  steam  raised 


256  WORK  OF  EXPANSION  OF  STEAM. 

from  a  cubic  foot  of  water.*     Since  the  volume  is  always  a 
function  of  the  pressure,  we  may  write 

V=f{F).  (1) 

136.  Work  of  Expansion  of  Steam. — Let  V  be  the 

volume  of  steam  from  a  cubic  foot  of  water  at  the  pressure 
P,  where  P  is  the  pressure  on  a  square  foot,  and  Uq  the 
work  performed  by  Q  cubic  feet  of  water,  in  the  form  of 
steam,  between  the  pressures  P  and  P^ ; 
let  ABCD  be  a  vertical  section  of  the 
space    in  which    the    steam    expands, 
ABPQ  the  volume  V  of  the  steam  at  the 
pressure  P,  ABP^Qj  the  volume  Fj  of 
the  steam  at  pressure  Pj,  A  the  area  of  F;g_  gg 

the  section  at  PQ  in  feet,  and  dv  the  dis- 
tance between  the  two  consecutive  sections  PQ  and  MN. 
Then,  for  the  element  of  work  performed  by  one  cubic  foot 
of  water  in  the  form  of  steam  at  the  pressure  P,  we  have 

dU^  =  KVdv  =  PdV, 

since  Adv  =  dV.     Integrating  between  the  limits  P  and 
Pj^,  we  have 

u^=  rpdv=  rpdf{P)       (1) 

[from  (1)  of  Art.  135]. 

Therefore,  for  the  work  done  by  Q  cubic  feet  of  water  in 
expanding  from  P  to  P ^,  we  have 

^q  =  Qf^Pdf{P),  (2) 

which  can  be  integrated  when  the  function  f{P)  is  known. 

ScH. — Since  from  (2)  the  quantity  of  work  done  is  en- 
tirely independent  of  the  form  of  the  vessel  ABCD,  it 

*  See  Ency.  Brit.,  Art.  Steam. 


WORK   OF  STEAM  AT  EFFLUX.  257 

follows  that  the  work  of  steam  between  any  given  pressures 
P  and  Pj  is  always  the  same,  whatever  may  be  the  nature 
of  the  space  through  which  the  steam  expands,  and  that  it 
increases  with  the  pressure  P  at  which  the  steam  is  gener- 
ated ;  since  tlierefore  the  quantity  of  fuel  necessary  to 
evaporate  a  given  volume  of  water  is  always  independent  of 
the  pressure  at  which,  the  steam  is  formed  (Art.  135),  it  fol- 
lows that  it  is  most  economical  to  employ  steam  of  as  high 
a  temperature  as  possible. 

137.    Work   of  Steam    at    Efflux.— Let  w  be  the 

weight  of  Q  cubic  feet  of  water  evaporated  per  second,  V 
the  volume  of  steam  from  a  cubic  foot  of  water  at  the  press- 
ure P,  which  is  the  pressure  of  the  steam  at  the  point  of 
efflux,  k  the  section  of  the  orifice  in  feet  from  which  the 
steam  is  discharged,  and  v  the  velocity  per  second.  Then 
calling  Uq  the  work  stored  in  the  steam  at  efflux,  we  have 

Since   V=.f{P),  and  QV  ^  kv,  we  have 
kv^Qf(P)  =  ^^nP); 

which  in  (1)  gives 

Hence,  the  work  of  steam  discharging  itself  from  an 
orifice  varies  as  the  cube  of  the  water  evaporated. 

Cor.  1. — The  work  stored  in  the  steam  at  efflux  is  due  to 
the  work  of  expansion  between  the  pressures  P  and  P^  ; 


258  WORK  OF  STEAM  AT  EFFLUX. 

therefore,  from  (2)  of  Art.  136  and  (1)  of  the  present  Art, 
we  have 

?/  =  "if'pviP)  =  ir^prnp) :     (4) 

which  gives  the  velocity  of  efflux,  fi  being  the  coefficient 
of  efflux  (Art.  131). 

Cor.  2. — From  (1)  of  Art.  130,  we  have 

trg  =  5M«  („^.  _  ,,»).  (6) 

Let  Fi  and  Fg  be  the  volumes  of  steam  per  second  from 
a  cubic  foot  of  water  at  P^  and  Pg  pressures  respectively ; 
then  (Art.  130), 

F  F 


a^  fltj 


which  in  (6)  gives 


(7) 


and  we  see  again,  as  in  (3),  that  the  worh  of  steam  varies 
as  the  cube  of  the  water  evaporated. 


Y 

Solving  (7)  for  Q  —^  =  v^,  we  have 


2^^e+nJ^iV 


/  F  \2     2 

_62.5(?    '    '^  Vof/  J  ' 
which  gives  the  theoretical  velocity  of  efflux. 


(8) 


WORK   OF  STEAM  Uf  THE  EXPANSIVE  ENGINE.     259 

138.  Work  of  Steam  in  the  Expansive  Engine. — 

Let  ^be  the  area  of  the  piston  in  square  feet,  hy  the  length 
of  the  stroke,  including  the  clearance,*  A  the  point  of  the 
cylinder  at  which  the  steam  is  cut  off,  Q  the  number  of 
cubic  feet  of  water  evaporated  per  minute,  P  and  P^  the 
pressures  of  the  steam  at  the  beginning  and  end  of  tlie 
stroke  respectively,  N  the  number  of  strokes  performed  by 
the  piston  per  minute,  V  the  volume  of  steam  from  a  cubic* 
foot  of  water  at  P  pressure,  and  V  =  f  {P),  as  before. 
Then,  for  the  volume  of  steam  discharged  per  minute  at  P 
pressure,  we  have 

Qf{P)  =  NKh.  (1) 

Similarly,  Qf{Pi)  =  NEh^y 

and  f{P,)  =  ^f{P).  (2) 

The  work  performed  upon  the  piston  before  the  steam  is 
cut  off  is  NKP  {h  —  c);  adding  this  to  (2)  of  Art.  136,  we 
have 

Uq=  Q  rPdf{P)  +  NKP{h  -  c) 

*^  Px 

^'prf/(P)+^-^P/(P)]        (3) 

[from  (1)], 
which  is  the  total  work  performed  hy  the  steam  per 
minute. 

Cor. — Let  L  be  the  useful  load  in  lbs.  upon  each  square 
foot  of  the  piston,  F  the  friction  in  lbs.  per  square  foot  of 
the  piston,  arising  from  the  motion  of  the  unloaded  piston, 
/the  coefficient  of  friction  arising  from  the  useful  load, and 

*  The  clearance  is  the  space  in  the  cylinder  lying  beneath  the  piston,  at  the  low> 
est  point  of  its  stroke. 


=<x: 


260  EXAMPLES. 

p  the  pressure  of  the  steam  in  the  condenser.  Then  the 
total  resistance  upon  the  piston  is  K  \^F  ^  L  {1  +  f)  4-  p], 
and  therefore  the  work  expended  per  minute  in  overcoming 
this  resistance 

=  NK[^F+L{1  +/)  +p]  (h,  -  c).  (4) 

When  the  mean  motion  of  the  piston  of  the  engine  is  uni- 
form, the  work  of  the  resistance  will  be  equal  to  the  work 
of  the  steam ;  therefore,  by  equating  (3)  and  (4),  and  re- 
ducing by  (1),  we  have 


h 


-f^Pdf{P)  +  {h-c)P 


f{Pyp. 

=  [F+L{l+f)+p-\{h,-c),       (5) 

from  which  the  value  of  the  useful  load  L  is  readily  de- 
termined. 

EXAM  PLES. 

1.  If  a  blowing  machine  changes  per  second  10  cubic  feet 
of  air,  at  a  pressure  of  28  inches,  into  a  blast  at  a  pressure 
of  30  inches,  find  the  work  to  be  done  in  each  second. 

Here^g,  from  (3a)  of  Art.  129,  =  0.49136  ;  .-.  etc. 

Ans.   1366.7  foot-lbs. 

2.  If  under  the  piston  of  a  steam  engine,  whose  area  is 
201  square  inches,  there  is  a  quantity  of  steam  15  inches 
high  and  at  a  pressure  of  3  atmospheres,  and  if  this  steam 
in  expanding  moves  the  piston  forward  25  inches,  find  the 
work  of  the  expansion  per  second.      Ans.  10866  foot-lbs. 

3.  The  air  in  a  reservoir  is  at  a  temperature  of  120°  C, 
and  at  a  pressure  corresponding  to  a  height  of  the  manom- 
eter of  5  inches,  while  the  barometer  marks  29.2  inches. 
Find  (1)  the  theoretical  velocity  of  efflux,  and  (2)  the  theo- 
retical discharge  through  an  orifice  \^  inches  in  diameter. 

Use  (9)  of  Art.  129. 

Ans.  (1)  645.12  feet ;  (2)  7.917  cubic  feet. 


EXAMPLES.  261 

4.  The  height  of  a  manometer,  which  is  placed  upon  a 
pipe  3 1  inches  in  diameter  through  which  the  air  is  passing, 
is  2^  inches,  while  the  air  is  discharged  th*^ough  an  orifice 

2  inches  in  diameter  at  the  end  of  the  pipe.  Find  (1)  the 
theoretical  velocity  of  efiiux,  and  (2)  the  theoretical  dis- 
charge, if  the  barometer  in  the  external  air  stands  at  27| 
inches,  and  the  air  in  the  pipe  is  at  a  temperature  of  10°  C. 

Use  (5)  of  Art.  132. 

Ans.   (1)  421.8  feet ;  (2)  9.2  cubic  feet. 

5.  If  in  the  last  example  the  height  of  the  manometer  is 

3  inches,  the  diameter  of  the  pipe  is  4  inches,  and  the  ori- 
fice at  the  end  of  the  pipe  is  1  inch  in  diameter,  find  (1)  the 
velocity,  and  (2)  the  discharge  when  the  barometer  stands 
at  29  inches  and  the  temperature  of  the  air  in  the  pipe  is 
20°  C.  Ans.   (1)  447.06  feet;  (2)  2.438  cubic  feet. 

6.  If  the  sum  of  the  areas  of  two  conical  tuyeres  of  a 
blowing  machine  is  3  square  inches,  the  temperature  in  the 
reservoir  is  15°,  the  height  of  the  manometer  in  the  regu- 
lator is  3  inches,  and  the  height  of  the  barometer  in  the 
exterior  air  is  20  inches,  find  the  discharge. 

See  (3)  of  Art.  132 ;  take  /^  =  .92,  and  a  =  .004.* 

Ans.  8.242  cubic  feet. 

7.  The  height  of  a  quicksilver  manometer,  which  is  placed 
upon  a  regulator  at  the  head  of  an  air  pipe  320  feet  long 
and  4  inches  in  diameter  is  3.1  inches,  the  heiglit  of  the 
barometer  in  the  free  air  is  29  inches,  the  diameter  of  the 
orifice  in  the  conically  convergent  end  of  the  pipe  is  2 
inches,  and  the  temperature  of  the  compressed  air  in  the 
regulator  is  20°  C.  Find  the  quantity  of  air  that  is  deliv- 
ered through  this  pipe. 

SXJG,     a  =  0.004,    //o  =  .75,    //j  =  .92;  .-.  etc. 

Ans.  5.735  cubic  feet. 

*  On  account  of  the  ordinary  humidity  of  the  atmosphere,  it  is  advisable  in  prac- 
tice to  take  a  =  0.004. 


262  EXAMPLES. 

8.  If  the  height  of  the  manometer  in  the  last  example  is 
4  inches,  the  pipe  500  feet  long  and  6  inches  in  diameter, 
the  height  of  the  barometer  30  inches,  the  diameter  of  the 
orifice  in  the  conically  convergent  end  of  the  pipe  2  inches, 
and  the  temperature  of  the  compressed  air  in  the  regulator 
30°  C,  find  the  quantity  discharged. 

Ans.  9.051  cubic  feet. 

9.  If  the  height  of  the  manometer  in  Ex.  7  is  2.5  inches, 
the  pipe  600  feet  long  and  5  inches  in  diameter,  the  height 
of  the  barometer  29.5  inches,  the  diameter  of  the  orifice  in 
the  conically  convergent  end  of  the  pipe  one  inch,  and  the 
temperature  of  the  compressed  air  in  the  regulator  is  10°  C, 
find  the  quantity  discharged.  Ans.  1.883  cubic  feet. 


CHAPTER    IV. 

HYDROSTATIC    AND    HYDRAULIC    MACHINES. 

139,  Definitions. — There  are  several  simple  machines 
whose  action  depends  on  the  properties  of  air  and  water  ;  a 
brief  description  of  some  of  these  machines  will  now  be 
given,  sufficient  to  exhibit  the  principles  involved  in  their 
construction  and  use. 

Hitherto  the  energy  exerted  by  means  of  a  head  of  water 
has  been  wholly  employed  in  overcoming  frictional  resist- 
ances, and  in  generating  the  velocity  with  which  the  water 
is  delivered  at  some  given  point.  In  the  cases  which  we 
have  now  to  consider,  only  a  fraction  of  the  head  is  required 
for  these  purposes;  the  remainder,  therefore,  becomes  a 
source  of  energy  at  the  point  of  delivery  by  means  of  which 
useful  work  may  be  done. 

Hydraulic  energy  may  exist  in  three  forms,  according  as 
it  is  due  to  motion,  elevation,  ov  pressure.  In  the  first  two 
cases  the  energy  is  inherent  in  the  water  itself,  being  a  con- 
sequence of  its  motion  or  position,  as  in  the  case  of  any 
other  heavy  body.  In  the  third  it  is  due  to  the  action  of 
gravity  or  some  other  force,  sometimes  on  the  water  itself, 
but  oftener  on  other  bodies  ;  the  water  then  only  transmits 
the  energy,  and  is  not  directly  the  source  of  it.* 

140.  The  Hydrostatic  Bellows. — This  machine  pre- 
sents an  illustration  of  the  principle  of  the  transmission  of 
fluid  pressure  (Art.  8).  It  consists  of  a  cylinder  CDEF 
(Fig.  70),  with  its  sides  made  of  leather  or  other  flexible 
material,  and  a  pipe  ABF  leading  into  it    If  water  is 

*  Cotterill's  App.  Meche.,  p.  488. 


264  THE  SIPHON. 

poured  into  the  pipe  till  the  vessel  and  pipe  are  filled,  a 
very  small  pressure  applied  at  A  will  raise  a  very  great 
weight  upon  DE,  the  weight   lifted  being 
greater  as  DE  is  greater. 

Let  h  be  the  area  of  a  horizontal  section  of 
the  pipe,  ^that  of  a  section  of  the  cylinder, 
or  that  of  DE,  and  p  the  pressure  applied  at 
A.     Then,  from  (1)  of  Art.  9,  we  have 

P  ^  /ix         \   ■■ 


ScH. — Suppose  the  pipe  AB  to  be  extend- 
ed vertically  upwards,  and  the  pressure  at  A  '^" 
to  be  produced  by  means  of  a  column  of  water  above  it, 
formed  by  pouring  in  water  to  a  considerable  height,  and 
suppose  the  pipe  to  be  very  small,  so  that  the  pressure  upon 
the  section  A  may  be  very  small ;  then,  as  this  pressure  is 
transmitted  to  every  portion  of  the  surface  DE  that  is  equal 
to  the  section  A,  the  upward  force  produced  on  DE  can  be 
as  large  as  we  please.  To  increase  the  upward  force,  we 
must  enlarge  the  surface  DE  or  increase  the  height  of  the 
column  of  water  in  the  pipe,  and  the  only  limitation  to  the 
increase  of  the  force  will  be  the  want  of  suflBcient  strength 
in  the  pipe  and  cylinder  to  resist  the  increased  pressure. 
By  making  the  pipe  AB  of  very  small  bore,  and  the  height 
DC  of  the  cylinder  very  small,  the  quantity  of  water  can  be 
made  as  small  as  we  please.  That  is,  any  quantity  of 
fluid,  however  small,  may  he  made  to  support  any 
weight,  however  great.  This  is  known  as  the  hydrostatic 
paradox. 

141.  The  Siphon. — The  action  of  a  siphon  is  an  im- 
portant practical  illustration  of  atmospheric  pressure.  It  is 
dimply  a  bent  tube  of  unequal  branches,  open  at  both  ends, 
and  is  used  to  convey  a  liquid  from  a  higher  to  a  lower 
level,  over  an  intermediate  point  higher  than  either. 


THE  SIPHON. 


265 


Fig.  71 


Let  A  and  B  be  two  vessels  containing 
water,  B  being  on  the  lower  level,  and 
ACB  a  bent  tube.  Suppose  this  tube  to 
be  filled  with  water  from  the  vessel  A, 
and  to  have  its  extremities  immersed  in 
the  water  in  the  two  vessels.  The  water 
will  then  flow  from  the  vessel  A  to  B,  as 
long  as  the  level  B  is  below  A,  and  the 
end  of  the  shorter  branch  of  the  siphon  is 
below  the  surface  of  the  water  in  the 
vessel  A. 

The  atmospheric  pressures  upon  the  surfaces  A  and  B 
tend  to  force  the  water  up  the  two  branches  of  the  tube. 
When  the  siphon  is  filled  with  water,  each  of  these  pressures 
is  counteracted  in  part  by  the  pressure  of  the  water  in  the 
branch  of  the  siphon  that  is  immersed  in  the  water  upon 
which  the  pressure  is  exerted.  The  atmospheric  pressures 
are  very  nearly  the  same  for  a  difference  of  level  of  several 
feet,  owing  to  the  slight  density  of  air.  The  pressures  of 
the  suspended  columns  of  water,  however,  will  for  the  same 
difference  of  level  differ  considerably,  in  consequence  of  the 
greater  density  of  water.  The  atmospheric  pressure  opposed 
to  the  weight  of  the  longer  column  will  therefore  be  more 
resisted  than  that  opposed  to  the  weight  of  ydie  shorter, 
thereby  leaving  an  excess  of  pressure  at  the  end  of  the 
shorter  branch,  which  will  produce  the  motion.  Thus, 
draw  the  vertical  line  DEC,  let  h  denote  the  height  of  the 
water  barometer,  k  the  area  of  a  section  of  the  tube,  and  w 
the  weight  of  a  unit  of  volume ;  then  the  water  at  the  point 
C  is  urged  from  left  to  right  by  a  force 


wTch  —  wk  X  EC 


and  it  is  urged  from  right  to  left  by  a  force 
=^  wkh  —  v'k  X  DC. 


266  THE  DIVING  'BELL. 

Subtracting  the  second  from  the  first,  we  have 

luk  (DC  —  EC)  =  wk  X  DE, 

for  the  resultant  force  which  urges  the  water  at  C  from  left 
to  right,  and  hence  there  will  be  a  continuous  flow  of  water 
from  the  upper  to  the  lower  vessel. 

It  will  be  observed  that  the  direction  of  the  flow  is  wholly 
due  to  the  fact  that  the  level  of  the  water  in  B  is  below  that 
of  the  water  in  A.  It  is  not  necessary  therefore  that  the 
longer  branch  should  be  immersed  in  the  water;  so  long 
as  the  end  B  of  the  tube  is  below  the  water  surface  in  A, 
the  water  will  continue  to  flow  through  the  tube  ACB, 
until  either  the  surface  in  A  has  fallen  below  the  end  of  the 
tube,  or,  if  the  siphon  be  long  enough,  until  the  surface  in . 
A  has  descended  so  far  that  its  depth  below  C  is  greater 
than  h. 

ScH. — The  siphon  is  often  used  to  drain  ponds,  marshes, 
and  canals,  and  when  used  for  this  purpose  it  is  made  of 
leather,  or  stout  canvas,  like  the  common  hose. 

142.  The  Diving  Bell. — This  is  a  large  bell-shaped 
vessel  made  of  iron,  open  at  the  bottom,  and  containing 
seats  for  several  persons.  Its  weight  is  greater  than  that  of 
the  water  it  would  contain,  and  when  lowered  by  a  chain 
into  the  water,  the  air  which  it  contains  becomes  more  and 
more  compressed  as  it  sinks,  in  consequence  of  the  increas- 
ing pressure  to  which  it  is  subject.  As  the  volume  of  air 
diminishes  the  water  rises  in  the  bell ;  but  the  air  will 
prevent  the  water  from  rising  high  in  the  bell,  and  the 
persons  seated  within  are  thus  enabled  to  descend  to  con- 
siderable depths  and  to  carry  on  their  operations  in  safety. 
When  the  surface  of  the  water  within  the  bell  is  at  a  depth 
of  33  feet  below  the  outer  surface  the  bell  will  be  half  filled 
with  water.  The  bell  is  supplied  with  fresh  air  from  above 
by  a  flexible  tube  connected  with  an  air  pump,  and  may  be 


THE  DIVING   BELL. 


267 


entirely  emptied  of  water  by  the  air  forced  in  by  the  purap. 
There  are  also  contrivances  for  the  expulsion  of  the  air  when 
it  becomes  impure. 

The  force  tending  to  lift  the  bell  is  the  weight  of  the 
water  displaced  by  the  bell  and  the  enclosed  air.  Hence 
the  tension  on  the  suspending  chain,  being  equal  to  the 
weight  of  the  bell  diminished  by  the  weight  of  water  dis- 
placed by  the  bell  and  the  air  within,  will  increase  as  the 
bell  descends,  in  virtue  of  the  diminution  of  air  space  due 
to  the  increased  pressure,  unless  fresh  air  is  forced  in  from 
above. 

Let  ABCD  be  the  bell,  let  EF  = 
a,  the  depth  of  its  top  below  the 
surface  of  the  water,  FK  =  h,  the 
height  of  the  cylinder,  FH  =  x,  the 
length  occupied  by  air,  tt  and  n'  the 
pressures  of  the  atmospheric  air  and 
of  the  compressed  air  within  the  bell, 
and  A  the  height  of  the  water  ba- 
rometer.    Then  we  have  (Art.  48) 


Fig.  72 


Tt'    =    TT-Z=    TT   ^   (jp.{a   +  X). 

But,  77  =  gph, 

which  in  (1)  gives 

x^  -{-  [a  +  h)  X  =^  hh, 


(1) 


_  -  (g  +  A)  +  a/  (<?  +  hf  -h  4M 

.     .       X   —  q  ,  \<<i) 

the  positive  value  only  being  the  one  which  belongs  to  the 
problem. 

Cor. — If  A  be  the  area  of  the  top  of  the  bell,  and  its 
thickness  be  neglected,  the  volume  of  displaced  water  is  A.x, 
and  the  tension  of  the  chain 

:z=  weight  of  bell  —  gpAjx,  (3) 


268 


THE   COMMON  PUMP. 


ScH. — The  principle  of  the  diving  bell  is  applied  in  div- 
ing dresses.  The  diver  is  clothed  in  a  water-tight  dress 
fitted  with  a  helmet,  and  is  suppUed  with  air  by  means  of  a 
pump.  There  is  an  escape  valve  bv  which  the  circulation 
of  fresh  air  is  maintained.  The  diver  may  be  weighted  up 
to  200  lbs.,  but  on  closing  the  escape  valve,  lie  can  rise  at 
once  to  the  surface  in  virtue  of  the  buoyancy  due  to  the 
increased  displacement  of  water  by  the  enclosed  air. 

143.   The  Common  Pump  (Suction  Pump).— Any 

machine*  used  for  raising  water  from  one  level  to  a  higher, 
in  which  the  agency  of  atmospheric  pressure  is  employed,  is 
called  a  pump.  Pumps  are  either  suction,  forcing,  or  lift- 
ing pumps. 

The  pump  most  commonly  in  use  is  a 
suction  pump,  of  which  Fig.  73  is  a  ver- 
tical section.  AB  and  BC  are  two  cylin- 
ders connected  together  having  a  common 
axis ;  the  former  is  called  the  barrel  of  the 
pump  and  the  latter  the  suction  pipe ;  M 
is  a  piston  accurately  fitting  the  barrel, 
and  movable  up  and  down  through  the 
space  AB  by  means  of  a  vertical  rod  EV, 
connected  with  a  handle  or  lever  EF, 
which  turns  on  a  fulcrum  0 ;  in  the  piston 
is  a  valve  V  which  opens  uj)wards,  and  at 
the  top  of  the  suction  pipe  BC  is  another 
valve  V,  which  likewise  opens  upwards. 
S  is  a  spout  a  little  above  A,  and  C  is 
the  surface  of  the  water  in  which  the  loAver  part  of  the 
pump  is  immersed. 

To  explain  the  action  of  the  suction  pump,  suppose  the 
piston  M  to  be  at  B,  the  pump  filled  with  ordinary  atmos- 

♦  Machines  for  raising  water  have  been  known  from  very  early  ages,  and  the 
invention  of  the  common  pump  is  generally  ascribed  to  Ctesibius,  teacher  of  the 
celebrated  Hero  of  Alexandria  ;  but  the  true  theory  of  its  action  was  not  under- 
Btood  till  the  time  of  Galileo  and  Torricelli.    (See  Dcsclianel's  Nat.  Phil.,  p.  215.) 


Fig.  73 


THE   COMMOX  PUMP.  269 

pheric  air,  and  the  valves  Y  and  V  closed  by  their  own 
weight;  the  water  will  stand  at  the  same  level  C  both 
within  and  without  the  suction  pipe.  Now  raise  the  piston, 
the  air  in  BC  will  tend  by  its  elastic  force  to  occupy  the 
space  which  the  piston  leaves  void;  it  will  therefore  open 
the  valve  V,  and  will  pass  from  the  pipe  to  the  barrel,  its 
elasticity  diminishing  in  proportion  as  it  fills  a  larger  space. 
It  will,  therefore,  exert  less  pressure  on  the  water  at  C 
than  the  atmosphere  does  at  C  outside  the  pump ;  hence 
the  atmospheric  pressure  on  the  surface  of  the  water  outside 
will  force  water  up  the  pipe  BC,  until  the  pressure  at  C  is 
equal  to  the  atmospheric  pressure.  As  the  piston  rises  the 
water  will  rise  in  BC,  the  pressure  of  the  air  above  M  keep- 
ing the  valve  V  closed.  When  the  piston  descends,  the 
valve  V  closes,  and  the  air  in  MB,  becoming  compressed  as 
the  piston  descends,  will  at  length  have  its  elastic  force 
greater  than  that  of  the  exterior  air  above  the  piston,  and 
will  open  the  valve  V,  and  will  escape  through  it. 

This  process  being  repeated  a  few  times,  the  water  at 
length  ascends  through  the  valve  V  into  the  barrel,  and  at 
the  next  descent  of  the  piston,  will  be  forced  through  the 
valve  V  and  be  then  lifted  to  the  spout  S,  through  which 
it  will  flow.  While  this  water  is  being  lifted,  the  atmos- 
pheric pressure  on  the  surface  of  the  water  outside  the  pipe 
forces  more  water  into  the  pump,  so  that,  on  the  next 
descent,  the  piston  gets  more  water  to  lift ;  and  thus  the 
process  continues,  the  suction  pipe  and  barrel  remaining 
full,  so  that  a  cylinder  of  water  equal  to  that  through  which 
the  piston  is  raised  will  be  poured  out  at  each  upward  mo- 
tion, provided  the  spout  S  is  large  enough. 

Sen.  1. — The  height  BC  must  be  less  than  the  height  of 
the  water  barometer,  or  else  the  water  will  never  rise  to  the 
valve  v.  Although  the  height  of  the  water  barometer  is 
about  33  feet,  yet  in  consequence  of  unavoidable  imperfec- 
tions in  construction,  the  height  of  the  Aalve  V  above  the 
surface  of  the  water  in  the  well  should  be  considerably  less 


270  TENSION   OF  THE  FISTON  ROD. 

than  33  feet;  otherwise  the  quantity  of  watep  lifted  by  the 
piston  at  each  stroke  will  be  small. 

ScH.  2. — It  is  not  essential  to  the  construction  that  there 
should  be  two  cylinders  ;  a  single  cylinder,  with  a  valve 
somewhere  below  the  lowest  point  of  the  piston -range  will 
be  sufficient,  provided  tlie  lowest  point  of  the  range  be  less 
than  33  feet  above  the  surface  in  the  reservoir. 

It  is  not  necessary  to  the  working  of  a  pump  that  the 
suction  pipe  should  be  straight;  it  may  be  of  any  shape, 
and  may  enter  the  reservoir  at  any  horizontal  distance  be- 
low the  barrel  of  the  pump. 

144.  Tension  of  the  Piston  Rod. — (1)  If  the  water  in 
BC  (Fig.  73)  has  risen  to  P  when  the  piston  is  at  M,  let  -n' 
be  the  pressure  of  the  air  in  MP  ;  then  we  have  -n'  = 
pressure  of  water  at  P  =  pressure  of  water  at  C  —  ^'pPC ; 
hence 

:r'  =  Tr-^pPC.  (1) 

But  the  tension  on  the  rod  is  the  difference  between  the 
atmospheric  pressure  above  the  piston  and  the  pressure  of 
the  air  in  MP  ;  hence  calling  A  the  area  of  the  piston  and 
T  the  tension  of  the  rod,  we  have  from  (1) 

T  =  (tt  _  tt')  A  =  ^pPC.  A.  (2) 

If  one  inch  be  taken  as  the  unit  of  length,  and  h  be  the 
height  in  inches  of  the  water  barometer,  we  have  gph  =  15 
lbs.,  nearly,  which  in  (2)  gives 

T  =  16^^.  (3) 

(2)  When  the  j)ump  is  in  full  action. — Let  AH  be 
the  range  of  the  piston,  and  let  CD  =  h,  then  at  each 
stroke,  the  volume  DH  of  water  is  lifted,  and  therefore  the 
tension  of  the  rod  when  the  piston  is  ascending  will  be 
gpA.  {h  +  HD)  until  the  water  begins  to  flow  through  the 
spout. 


HEIGHT   WHICH   WATER   RISES  IX  ONE  STROKE.    271 

Therefore  in  the  suction  pump  the  tension  of  the 
rod  is  equaZ  to  the  weight  of  the  colv/mn  of  water 
wJiose  base  is  the  area  of  the  piston  and  whose  height 
is  the  height  of  the  water  in  the  pump  above  the  level 
of  the  well. 

If  A  be  on  a  level  with  the  spout,  all  the  water  lifted 
will  be  discharged,  and  as  the  piston  descends,  the  tension 
of  the  rod  will  be  gpAli. 

145.  Height  Through  which  the  Water  Rises  in 
One  Piston  Stroke — Let  P  and  Q  (Fig.  73)  be  the  sur- 
faces of  the  water  at  the  beginning  and  end  of  an  upward 
stroke  of  the  piston  from  B  to  A,  and  let  h  as  usual  be  the 
height  of  the  water  barometer.  The  air  Avhich  occupied  the 
space  BP  at  the  beginning  of  the  stroke  occupies  at  the  end 
of  it  the  space  AQ ;  and  the  pressures  are  respectively 

gp{h~VQ),  gpih-qC). 

Hence  (Art.  48) 

h  —  FC:h  —  qC   : :   vol.  AQ  :  vol.  BP. 

If  B  and  r  are  the  radii  of  the  cylinders  AB  and  BC,  we 

have 

vol.  AQ  =  -B^AB  +  7rr2  (BC  —  QC), 

vol.  BP  =  7rr2  (BC  -  PC), 

^_PC  _  B^AB  +  yz  (BC  —  QC) 
"*•     ^_QC-  r2(BC-PC) 

which  determines  QC  for  any  given  value  of  PC. 

Cor. — If  the  stroke  of  the  piston  be  less  than  AB,  as  for 
instance  AH,  then  HC  must  be  less  than  h.  Also,  a  limit 
exists  with  regard  to  H,  which  may  be  shown  as  follows  : 

If  P  be  the  surface  of  the  Avater  when  the  piston  M  is  at 
A,  then,  as  the  piston  descends,  the  valve  V  will  close,  but 
the  valve  V  will  not  be  opened  until  the  pressure  of  the  air 


272   HEIGHT   WHICH   WATER   RISES  IJST  ONE  STROKE. 

in  MB  is  greater  than  the  atmospheric  pressure.  When 
M  is  at  A  the  pressure  of  the  air  =  gp{h  —  PC),  and  un- 
less the  valve  V  is  opened  before  M  arrives  at  H,  the  pressure 
of  the  air  in  HB  will 

AT? 

which  must  be  greater  than  gpit,  if  the  valve  is  to  open,  and 
therefore  A  "AH  must  be  greater  than  AB-PC.  Hence,  to 
insure  the  opening  of  the  valve  while  the  surface  is  below 
B,  we  must  have 

A-AH  >  AB-BO,  (1) 

AH  ^    BC 
AB  >  X  ' 

i.  e.,  the  ratio  of  AH  to  HB  must  be  at  least  as  great  as  the 
ratio  of  BC  to  7i.  This  condition,  although  necessary  in 
every  case,  may  not  be  sufficient. 

For,  suppose  that  the  surface  of  the  water  is  at  Q'  when 
the  piston  M  is  at  A,  in  which  case  the  pressure  of  the  air 
in  AQ'  =  gp  {7i  -  Q'C). 

When  the  piston  descends  to  H,  the  pressure  in  HQ' 

=  5rp(A_Q'C)|^|, 

which  must  be  greater  than  gp7i,  if  the  valve  is  to  open,  and 

therefore 

7«.AH  >  AQ'.QC 

But  the  greatest  value  of  AQ'-Q'C  is  JAC'^;  therefore 
we  must  have 

A-AH  >  iACl  (2) 

Since  ^A&  >  AB-BC,  unless  B  is  the  middle  point  of 
AC,  it  follows  that  the  condition  in  (2)  includes  the  condi- 
tion in  (1),  which  is  therefore  in  general  insufficient.  (See 
Besant's  Hydrostatics,  p.  97.) 


THE  LIFTING   PUMP. 


273 


Fig.  74 


146.  The  Lifting  Pump.  — AVhen  water  has  to  be 
raised  to  a  height  exceeding  about  30  feet,  the  suction  pump 
will  not  work  (Art.  143,  Sch.  1),  and 
the  lifting  pump  is  commonly  used. 
By  means  of  this  instrument,  Avater 
can  be  lifted  to  any  heiglit.  It  con- 
sists of  two  cylinders,  in  the  upper 
of  which  a  piston  M  is  movable,  the 
piston-rod  working  through  an  air- 
tight collar.  A  pipe  DF  is  carried 
from  the  barrel  to  any  required 
height;  at  D  there  is  a  valve  which 
opens  into  the  pipe.  The  suction 
pipe  BC  is  closed  by  a  valve  V,  as 
in  the  suction  pump,  and  the  piston 
M  usually  *  has  a  valve  V. 

The  action  of  this  pump  is  precisely  the  same  as  that  of 
the  suction  pump  in  raising  Avater  from  the  well  into  the 
barrel.  Suppose  the  piston  at  its  highest  point,  and  the 
surface  of  the  water  in  the  barrel  at  K  ;  then,  as  the  piston 
is  depressed,  its  valve  V  will  open,  and  the  water  will  flow 
through  it  till  the  piston  reaches  its  lowest  point.  When 
the  piston  ascends,  lifting  the  water,  the  valve  D  opens,  and 
water  ascends  in  the  pipe  DF.  On  the  descent  of  the  piston, 
the  valve  D  closes,  and  every  successive  stroke  increases  the 
quantity  of  water  in  the  pipe,  until  at  last  it  is  filled,  after 
which  every  elevation  of  the  piston  will  deliver  a  volume  of 
water  equal  to  that  of  a  cylinder  whose  base  is  the  area  of 
the  piston  and  whose  height  is  equal  to  its  stroke.  The 
only  limit  to  the  height  to  which  water  can  be  lifted  is  that 
which  depends  on  the  strength  of  the  instrument  and  the 
power  by  which  the  piston  is  raised. 

Cor. — If  CK  =  h,  the  piston  lifts  the  volume  BK  at 


*  Sometimes  the  piston  has  no  valve  in  it,  bat  is  replaced  by  a  solid  cylinder, 
called  a  plunger,  which  is  operated  by  a  handle  as  before. 


274 


TSE  FORCING   PUMP. 


each  stroke,  and  if  ^  =  the  area  of  the  piston,  the  tension 
on  the  piston-rod  =  gpA'  BK,  until  the  water  is  lifted  to 
the  valve  D,  since  the  air  is  expelled  before  the  machine  is 
in  full  action.  After  this,  the  power  applied  to  the  piston- 
rod  must  be  increased  until  the  pressure  of  the  water  opens 
the  valve  D,  i.  e.,  until  the  pressure  =5'p  {h  +  DF),  where 
F  is  the  surface  of  the  water  in  the  tube.  The  water  will 
then  be  forced  up  the  tube,  the  tension  of  the  rod  increas- 
ing as  the  surface  F  ascends.  ' 


147.  The  Forcing  Pump.— This  pump  is  a  further 
modification  of  the  simple  suction  pump;  it  has  no  valve  in 
its  piston,  which  is  perfectly  solid, 
and  works  water-tight  in  the  barrel, 
ranging  over  the  space  AE.  At  the 
top  of  the  suction  pipe  BC  is  a  valve, 
and  at  the  entrance  to  the  pipe  DF  is 
a  second  valve  D. 

When  this  pump  is  first  set  in  ac- 
tion, water  is  raised  from  the  well  as 
in  the  common  pump,  by  means  of 
the  valve  B  and  piston  M,  the  air  at 
each  descent  of  the  piston  being 
driven  through  the  valve  D  into  the 
pipe  DF.  When  the  water  has  risen 
through   B,  the   piston,  descending, 

forces  it  through  D  ;  and  when  the  piston  ascends,  the  valve 
D  closes,  and  more  water  enters  through  B.  The  next  de- 
scent of  the  piston  forces  more  water  through  D,  and  so  on 
until  the  pipe  is  filled,  as  in  the  lifting  pump. 

The  stream  which  flows  from  the  top  of  the  pipe  will  be 
intermittent,  as  it  is  only  on  the  descent  of  the  piston  that 
water  is  forced  into  the  pipe  ;  but  a  continuous  stream  can 
be  obtained  by  means  of  a  strong  air  vessel  N  (Fig.  76), 
which  consists  of  a  strong  brass  or  copper  vessel,  at  the  bot- 
tom of  which  is  a  valve  V.     Through  the  top  of  the  air 


Fig.  75 


THE  FORCING   PUMP. 


275 


Fig.  76 


vessel  is  a  discharge  pipe  KF,  which  passes  air-tight  nearly 
to  the  bottom.  When  water  is  forced  into  the  air  vessel 
through  the  valve  V  by  the  de- 
scent of  the  piston,  it  rises 
above  the  lower  end  of  this 
pipe.  The  mass  of  air  which 
the  vessel  contains  is  compressed 
into  a  smaller  volume ;  its  elas- 
tic force,  pressing  on  the  sur- 
face of  the  water  at  K,  with  a 
varying  but  continuous  press- 
ure, forces  it  up  the  pipe ;  and 
if  the  size  of  the  vessel  be  suit- 
able to  that  of  the  pump,  and  to 
the  rate  of  working  it,  the  com- 
pressed air  will  continue  to  ex- 
pand, forcing  water  up  the  pipe  during  the  ascent  of  the 
piston,  and  will  not  have  lost  its  force  before  a  new  com- 
pression is  applied  to  it,  carrying  with  it  a  new  supply  of 
water,  and  thus  a  continuous,  although  varying,  flow  will 
be  maintained.  A  few  stroke?  of  the  piston  will  generally 
be  sufficient  to  raise  water  in  the  pipe  KF,  to  any  height 
consistent  with  the  strength  of  the  instrument  and  the 
power  at  command. 

CoK. — Let  h  =  the  height  of  the  water  barometer ;  dur- 
ing the  ascent  of  the  piston  the  valve  B  is  open  and  V  is 
closed ;  the  pressure  upon  the  upper  surface  of  the  piston  = 
gph ;  the  pressure  upon  the  lower  surface  =  gp  {h  —  MC), 
the  water  surface  in  the  pump  being  at  M ;  therefore,  call- 
ing A  the  area  of  the  piston,  the  tension  of  the  rod  when 
the  piston  is  ascending  =  gpA  •  MC. 

That  is,  the  tension  of  the  rod  is  equal  to  the  weight 
of  a  column  of  ivater  ivhose  base  is  the  area  of  the 
piston,  and  ivhose  height  is  the  height  of  the  water  in 
the  barrel  above  the  level  of  the  well. 


276 


bramah's  press. 


U8.  The  Fire  Engine.— This  is  only  a  modification 
of  the  forcing  pump  with  an  air 

vessel,  as  fust  described. 

Two  cylinders  M  and  M'  are 
connected  with  the  air  vessel  V 
by  means  of  the  valves  D  and 
D',  and  the  pistons  are  worked 
by  means  of  a  lever  GEG',  the 
ends  of  which  are  raised  and  de- 
pressed alternately,  so  that  one 
piston  is  ascending  while  the 
other  is  descending.  Water  is  thus  continually  being 
forced  out  of  the  air  vessel  through  the  vertical  pipe  EH, 
which  has  a  flexible  tube  of  leather  attached  to  it,  by  means 
of  which  the  stream  can  be  thrown  in  anv  direction. 


149.  Braniah's  Press.*— This  press  is  a  practical  ap- 
plication of  the  principle  of  the  equal  transmission  of  fluid 
pressures  (Art.  8).  In  the  vertical 
section  of  this  instrument  (Fig. 
78),  A  and  C  are  two  solid  pistons 
or  cylinders  fitting  in  air-tight  col- 
lars, and  working  in  the  strong 
hollow  cyhnders  L  and  K,  which 
are  connected  by  a  pipe  BD.  At 
D  is  a  valve  opening  upwards,  arid 
at  B  is  a  valve  opening  inwards,  a 
pipe  from  D  communicating  with 
a  reservoir  of  water.  M  is  a  mova- 
ble platform,  supporting  the  substance  to  be  pressed,  and  N 
is  the  top  of  a  strong  frame.  HOF  is  the  lever  working 
the  cylinder  C,  F  being  the  fulcrum,  and  H  the  handle. 

Action  of  the  Press. — Let  C  be  raised ;  the  atmospheric 

♦  The  principle  of  this  press  was  sugsfested  by  Stevinus.  It  remained  unfruitful 
in  practice  until  1796,  when  Bramnh.  an  Gnorlisb  engineer,  by  an  ingenious  con- 
trivance, overcame  the  only  diflSculty  which  prevented  its  practical  application. 


Fig.  78 


HAWK8BEE  S  AIR-PUMP. 


277 


pressure  forces  water  from  the  reservoir  through  the  valve 
D  into  the  hollow  cyhnder  K,  as  in  the  comujon  pump. 
The  cyliuder  C  being  pressed  down,  the  valve  D  closes,  and 
the  water  is  forced  through  the  valve  B  into  L,  and,  acting 
on  the  cylinder  A,  makes  it  ascend,  thus  producing  pressure 
upon  any  substance  included  between  M  and  N.  A  con- 
tinued repetition  of  this  process  will  produce  any  required 
compression  of  the  substance. 

Let  R  and  r  be  the  radii  of  the  cylinders  A  and  C,  p  the 
power  applied  at  the  handle  H,  and  P  the  pressure  of  the 
water  on  A  ;  then  we  have,  for  the  downward  force  j?'  on  C, 


P   =P 


HF 
OF* 


But  (Art.  9) 


P-.p'  =  R^'.r^', 
HF  ^ 


(1) 


(2) 


By  increasing  the  ratio  of  R  to  r,  any  amount  of  pressure 
may  be  produced.  Presses  of  this  kind  were  employed  in 
lifting  into  its  place  the  Britannia  Bridge  over  the  Menai 
Straits,  and  for  launching  the  Great  Eastern. 


150.  Hawksbee's  Air-Pump.*— Band  B' are  two  cylin- 
ders, in  which  pistons  P  and  P',  with  valves  V  and  V  opening 
upward,  are  worked  by  means  of 
a  toothed  wheel,  the  one  ascend- 
ing as  the  other  descends.  At  the 
lower  extremity  of  the  cylinders 
there  are  valves  v  and  v'  opening 
upwards,  and  communicating  by 
means  of  the  pipe  AC  with  the 
receiver  R,  from  which  the  air  is 
to  be  exhausted.  Fig.  79 


*  The  air-pump  was  invented  in  1650  by  Otto  von  Guericke,  Burgomaster  of 
Magdeburg. 


278  HAWKSBEE^S  AIR-PUMP. 

Suppose  P  at  its  lowest  and  P'  at  its  highest  position, 
and  turn  the  wheel  so  that  P  ascends  and  P'  descends. 
When  P'  descends,  the  valve  v'  closes  and  the  air  in  B'  flows 
through  V",  while  the  valve  V  is  closed  by  the  pressure  of 
the  external  air,  and  air  from  R,  by  its  elastic  force,  opens 
the  valve  v  and  fills  the  cylinder  B.  When  P  descends,  the 
valve  V  closes,  and  the  air  in  B  being  compressed  flows 
through  the  valve  V,  while  the  valve  V  closes,  and  air  from 
the  receiver  flows  through  v'  into  B'.  At  every  stroke  of 
the  piston,  a  portion  of  the  air  in  the  receiver  is  withdrawn  ; 
and  after  a  considerable  number  of  strokes  a  degree  of  rare- 
faction is  attained,  which  is  limited  only  by  the  weight  of 
the  valves  which  must  be  lifted  by  the  pressure  of  the  air 
beneath. 

Let  A  denote  the  volume  of  the  receiver,  and  B  that  of 
either  cylinder ;  p  the  density  of  atmospheric  air,  and  pj, 

Pg, pn  the  densities  in  tlie  receiver  after  1,  2, n 

descents  of  the  pistons.  Then  after  the  first  stroke  the  air 
which  occupied  the  space  A  will  occupy  the  space  A  +  B, 
and  therefore  we  have 

Pi  {A  +  B)  =  pA. 

Similarly,  p^  {A  -}-  B)  =  p^A  ; 

.-.    p^{A-{-Bf  =  pA% 

and  after  n  strokes  we  have 

Pn  (A  +  B)^  =  pA^, 

the  volume  of  the  connecting  pipe  AC  being  neglected. 

Hence,  calling  7r„  and  tt  the  pressures  of  the  air  in  the 
receiver  after  n  strokes  and  of  the  atmospheric  air  respect- 
ively, we  have 

^  _Pn  _  /_4_\"  n\ 

n   -  p  -  \A  +  b)'  ^  ' 

Thus,  suppose  that  A  is  foar  times  B,  and  we  were  re- 


SMEATON  S  AIR-PUMP, 


279 


quired  to  find  the  density  of  the  air  in  the  receiver  at  the 
end  of  the  15th  stroke,  \ve  have  from  (1) 

Pi  5  =p(i)^  =  0.03552  p. 

If  the  ail-  originally  had  an  elastic  force  equal  to  the 
pressure  of  30  in.  of  mercury,  this  would  give  the  elastic 
force  of  the  air  remaining  in  the  receivjer  as  equal  to  a 
pressure  of  1.056  in.  of  mercury.  In  this  case,  it  is  custom- 
ary to  say  that  the  vacuum  pressure  is  one  of  1.056  in.  of 
mercury. 

ScH. — It  is  evident  from  (1)  that  pn  can  never  become 
zero  as  long  as  n  is  finite,  and  therefore,  even  if  the  machine 
were  mechanically  perfect,  we  could  not  by  any  number  of 
strokes  completely  remove  the  air;  for,  after  every  stroke 
there  would  be  a  certain  fraction  left  of  that  which  occupied 
it  before. 

In  working  the  instrument,  the  force  required  is  that 
which  will  overcome  the  friction,  together  with  the  differ- 
ence of  the  pressures  on  the  under  surfaces  of  the  pistons, 
the  pressures  on  their  upper  surfaces  being  the  same. 

151.  Smeaton's  Air-Pump. — This  instrument  con- 
sists of  a  cylinder  AB  in  which  a  piston  is  worked  by  a  rod 
passing  through  an  air-tight  collar  at  the 
top ;  a  pipe  BD  passes  from  B  to  the 
glass  receiver  C,  and  three  valves,  open- 
ing upwards,  are  placed  at  B,  A,  and  in 
the  piston. 

Suppose  the  receiver  and  cylinder  to 
be  filled  with  atmospheric  air,  and  the 
piston  at  B.  Raising  the  piston,  the 
valve  A  is  opened  by  the  compressed  air 
in  AM  which  flows  out  through  it,  while  at  the  same  time 
a  portion  of  the  air  in  C  flows  through  the  pipe  DB  to  fill 
the  partial  vacuum  formed  in  MB,  so  that  when  the  piston 
arrives  at  A,  the  air  which  at  first  occupied  C  now  fills  both 


f=i 


&J 


Fig.  80 


;880 


THE  HYDRAULIC  HAM. 


the  receiver  and  the  cylinder.  When  the  piston  descends, 
the  valves  A  and  B  close,  and  the  air  in  the  cylinder  below 
the  piston  is  compressed  until  it  opens  the  valve  M,  and 
passes  above  the  piston.  As  the  piston  is  raised  a  second 
time  the  valve  A  is  opened  by  the  compressed  air  in  AM, 
which  flows  out  through  it  as  before;  and  thus  at  each 
stroke  of  the  piston  a  portion  of  the  air  in  the  receiver  is 
forced  out  through  A. 

Let  A  and  B  denote  the  volumes  of  the  receiver  and 
cylinder  respectively,  and  p  and  pn  the  densities  of  atmos- 
pheric air  and  of  air  in  the  receiver  after  n  strokes.  Then, 
as  in  Art.  150,  we  have 

Pn  {A  +  i?)«  =  pA^, 
from  which   it  appears  as  in    the  previous  article   that, 
although  the  density  of  the  air  will  become  less  and  less 
at  every  stroke,  yet  it  can  never  be  reduced  to  nothing, 
however  great  n  may  be. 

ScH. — An  advantage  of  this  instrument  is  that,  the  upper 
end  of  the  cylinder  being  closed,  when  the  piston  descends 
the  valve  A  is  closed  by  the  external  pressure,  and  therefore 
the  valve  M  is  then  opened  easily  by  the  air  beneath.  Also 
the  labor  of  working  the  piston  is  diminished  by  the 
removal,  during  the  greater  part  of  the  stroke,  of  the 
atmospheric  pressure  on  M,  which  is  exerted  only  during 
the  latter  part  of  the  ascent  of  the  piston,  when  the  valve 
A  is  open. 

152.  The  Hydraulic 
Ram.* — The  hydraulic  ram 
is  a  machine  by  which  a  fall 
of  water  from  a  small  height 
produces  a  momentum  which 
is  made  to  force  a  portion  of 
the  water  to  a  much  greater 
height.  Fifl.  81 


*  Invented  by  Montgolfier. 


THE  HYDRAULIC  RAM.  281 

111  the  vertical  section  (Fig.  81),  AB  is  the  descending 
and  FG  the  ascending  column  of  water,  which  is  sup- 
plied from  a  reservoir  at  A.  V  is  a  valve  opening  down- 
wards, and  V  is  a  valve  opening  upwards  into  the  air- 
vessel  C ;  H  is  a  small  auxiliary  air-vessel  with  a  valve  K 
opening  inwards. 

Ttie  Action  of  the  Machine. — As  the  valve  V  at  first 
lies  open  by  its  own  weight,  a  portion  of  the  water,  descend- 
ing from  A,  flows  through  it ;  but  the  upward  flow  of  the 
water  towards  the  valve  V  increases  the  pressure  tending  to 
lift  the  valve,  and  at  last,  if  the  valve  is  not  too  heavy,  lifts 
and  closes  it.  The  forward  momentum  of  the  column  of 
water  ABD  being  destroyed  by  the  stoppage  of  the  flow, 
the  water  exerts  a  pressure  sufficient  to  open  the  valve  V 
and  to  flow  through  it  into  the  air-vessel  C,  condensing  the 
air  within ;  the  reaction  of  the  condensed  air  forces  water 
up  the  pipe  FG.  As  the  column  of  water  ABD  comes  to 
rest,  the  pressure  of  the  water  diminishes,  and  the  valves 
V  and  V  both  fall.  The  fall  of  tiie  former  produces  a 
rush  of  the  water  through  the  opening  V,  followed  by  an 
increased  flow  down  tlie  supply  pipe  AB,  the  result  of  which 
is  again  the  closing  of  V,  and  a  repetition  of  the  process 
just  described,  the  water  ascending  higher  in  FG,  and 
finally  flowing  through  G. 

The  action  of  the  machine  is  assisted  by  the  air-vessel  H 
in  two  ways — first,  by  the  reaction  of  the  air  in  H,  which  is 
compressed  by  the  descending  water,  and,  secondly,  by  the 
valve  K,  which  affords  supplies  of  fresh  air.  When  the 
water  rises  through  V,  the  air  in  H  suddenly  expands,  and 
its  pressure  becoming  less  than  that  of  the  outer  air,  the 
valve  K  opens,  and  a  supply  flows  in,  which  compensates 
for  the  loss  of  the  air  absorbed  by  the  water  and  taken  up 
the  column  FG,  or  wasted  through  V.  About  a  third  of 
the  water  employed  is  wasted,  but  the  machine  once  set  in 
motion  will  continue  in  action  for  a  long  time,  provided  the 


282  WORK   OF  WATER    WHEELS. 

supply  in  the  reservoir  be  maintained.  (See  Besant's 
Hydrostatics,  p.  112.) 

153.  Work  of  Water  Wheels. — To  utilize  ahead  of 
water,  consisting  of  an  actual  elevation  above  a  datum  level 
at  which  the  water  can  be  delivered  and  disposed  of,  a 
machine  may  be  employed  in  which  the  direct  action  of 
the  weight  of  the  water,  while  falhng  through  the  given 
height  is  the  principal  moving  force. 

When  a  stream  of  water  strikes  the  paddles  of  a  wheel 
which  has  a  certain  velocity,  the  energy  imparted  to  the 
wheel  by  the  water,  from  (4)  of  Art.  98, 

=  [v'-(v-VY]^^,  (1) 

where  F  is  the  velocity  of  the  periphery  of  the  wheel,  v  the 
original  velocity  of  the  water,  and  W  the  weight  of  water 
acting  on  the  wheel  per  second ;  but  if  the  water  descends 
with  the  paddle  there  is  an  additional  amount  of  work  done 
on  the  wheel  due  to  the  mean  height  h  through  which  the 
water  falls.  Hence  we  have,  for  the  whole  work  done  on 
the  wheel  per  second, 

=  [^,2  _  (i,  _  vy]  ^  +  WJi.  (2) 

Now  if  the  water  leaves  the  paddles  the  work  remaining 
in  the  water  will  be  lost ;  hence,  calling  v^  the  velocity  of 
the  water  after  it  has  left  the  paddles,  we  have  for  the  use- 
ful work  U  done  on  the  wheel 

ir=  [i;2  _  (y  _  F)2  -  f  ,2]  ^  +  ^A 

W 
=  l2vV-V^-v,^]^  +  Wh,  (3) 

which  is  the  general  expression  for  the  work  done  by  a 
water  wheel  when  the  water  impinges  upon  the  paddles 
perpendicularly. 


WORK   OF  OVERSHOT   ffHEELS. 


283 


f^i^'/'82 


154.  Work  of  Overshot  Wheels. — When  a  waterfall 
ranges  between  10  and  70  feet,  and  the  water  supply  is  from 
3  to  25  cubic  feet  per  second,  it  is 
possible  to  construct  a  bucket 
wheel  on  which  the  water  acts 
chiefly  by  its  weight.  If  the  varia- 
tion of  the  head-water  level  does 
not  exceed  2  feet,  an  overshot 
wheel  may  be  used.  The  water  is 
then  projected  over  the  summit  of 
the  wheel,  and  falls  in  a  parabolic 
path  into  the  bucket.  If  v  be  the 
velocity  of  delivery  to  the  wheel, 

the  part  x-  is  converted  into  energy  of  motion  before  reach- 
ing the  buckets  and  operates  by  impulse  ;■  hence  in  a  wheel 
of  this  class  the  water  does  not  operate  entirely  by  weight. 

The  height  h  through  which  the  water  falls  is  the  vertical 
height  of  the  point  at  which  the  water  meets  the  buckets 
above  the  point  where  it  leaves  them,  which  in  this  wheel 
is  nearly  equal  to  the  diameter  of  the  wheel ;  and  as  the 
velocity  of  the  water  on  leaving  the  bucket  is  the  same  as 
the  velocity  of  the  bucket  itself,  we  have  v^  =V;  hence 
(3)  of  Art.  153  becomes 

U=  {v-V)V~  +  Wh.  (1) 


Calling  m  the  efficiency*  of  these  wheels,  we  have  from  (1) 


U=  m 


±  (y  _  F)  F  4-  h 


W. 


(2) 


Cor. — To  find  the  relation  of  v  and  V  so  that  the  useful 
work  U  of  the  wheel  may  be  a  maximum,  we  must  equate 
to  zero  the  derivative  of  ZJwith  respect  to  F,  which  gives 


♦  See  Anal.  Mechs,,  Art.  316. 


284  WORK   OF  BR  Ji A  ST   WHEELS. 

V  =:  ^v,  i.  e.,  the  ivheel  works  to  the  best  advantage 
when  the  velocity  of  its  periphery  is  one-half  that  of 
the  stream. 

ScH. — If  the  velocity  of  the  periphery  of  this  wheel  is 
too  great,  water  is  thrown  out  of  the  buckets  before  reach- 
ing the  bottom  of  the  fall.  In  practice,  the  circumferential 
velocity  of  Avater  wheels  of  this  kind  is  from  4^  to  10  feet 
per  second,  about  6  feet  being  the  usual  velocity  of  good 
iron  wheels  not  of  very  small  size.  The  velocity  of  the 
water  therefore  is  limited  to  about  13  feet  per  second,  and 
the  part  of  the  fall  operating  by  impulse  is  therefore  about 
2J  feet.  The  rest  of  the  fall  operates  by  gravitation,  but  a 
certain  fraction  is  wasted  by  spilling  from  the  buekets,  and 
emptying  them  before  reaching  the  bottom  of  the  fall.  The 
great  diameter  of  .wheel  required  for  very  high  falls  is  in- 
convenient, but  there  are  examples  of  wheels  60  feet  in 
diameter  and  more. 

The  efficiency  of  these  wheels  under  favorable  circum- 
stances is  0.75,  and  is  generally  about  0.65.. 

155.  Work  of  Breast  Wheels. — When  the  variation 
of  the  head-water  level  exceeds  2  feet,  a  breast  wheel  is 
better  than  an  overshot.  In 
breast  wheels  the  buckets  are 
replaced  by  vanes  which  move  in 
a  channel  of  masonry  partially 
surrounding  the  wheel.  The 
water  falls  over  the  top  of  a  slid- 
ing sluice  in  the  upper  part  of 
the  channel.  The  channel  is 
thus  filled  with  water,  the  weight  -.^.^,^,■^■^■■^^■ 

of  which  rests  on  the  vanes  and 
furnishes  the  motive  force  on  the  wheel.  There  is  a  certain 
amount  of  leakage  between  the  vanes  and  the  sides  of  the 
channel,  but  this  loss  is  not  so  great  as  that  by  spilling  from 
the  buckets  of  the  overshot  wheel, 


W^ORK  OF  UNDERSHOT   WHEELS. 


285 


In  this  wheel,  as  in  the  case  of  the  overshot  wheel,  v^  = 
V,  therefore  (1)  and  {2)  of  Art.  154  also  apply  to  breast 
wheels,  h  being  the  height  of  the  point  at  which  the  water 
meets  the  vanes  above  the  point  wiiere  it  leaves  them.  The 
efficiency  is  found  by  experience  to  be  as  much  as  0.75. 

ScH.  1. — Theoretically  this  wheel  also  works  to  the  best 
advantage  when  the  speed  of  its  periphery  is  one-half  that 
of  the  stream  (Art.  154,  Cor.).  But  Morin  found,  by  ex- 
periments, that  the  etficiency  of  the  wheel  is  not  much 
affected  by  changes  in  its  velocity.  This  is  owing  to  the 
circumstance  that  the  useful  work  is  dependent  principally 
upon  the  term  W/i,  and  not  upon  the  other  term  in  the 
formula  which  alone  is  affected  by  the  velocity  of  the  wheel. 
Hence  the  great  advantage  of  this  wheel  is,  that  it  may  be 
worked,  without  materially  impairing  its  efficiency,  with 
velocities  varying  from  ^v  to  f  v. 

ScH.  2. — As  the  diameter  of  this  wheel  is  greater  than 
the  fall,  a  breast  wheel  can  be  employed  only  for  moderate 
falls. 

Overshot  and  breast  wheels  work  badly  in  back-water, 
and  hence  if  the  tail-water  level  varies,  it  is  better  to  reduce 
the  diameter  of  the  wheel  so  that  its  greatest  immersion  in 
flood  is  not  more  than  one  foot. 


156.  Work  of  Undershot 
Wheels. — The  common  un- 
dershot wheel  consists  of  a 
wheel  provided  with  vanes, 
against  which  the  water  im- 
pinges directly.  In  this  case 
the  water  is  allowed  to  attain 
a  velocity  due  to  a  considera- 
ble part  of  the  head  immediately  before  entering  the  ma- 
chine, so  that  its  energy  is  nearly  all  converted  into  energy 
of  motion ;  and  as  the  water  has  no  fall  on  the  wheel,  and 


286  WOBK   OF  THE  PONCE  LET   WATER    WHEEL. 

its  velocity  on  leaving  the  vanes  is  the  same  as  the  velocity 
of  the  vane  itself,  we  have  h  =  0,  Vi  =  F;  therefore  (3) 
of  Art.  153  becomes 

U={v-V)V^,  (1) 

w 

or  .  U=  m(v—  V)  F— ,  (2) 

where  m,  as  before,  is  the  efficiency  of  the  machine. 

ScH. — The  wheel  works  to  the  best  advantage  when  the 
speed  of  the  periphery  is  one-half  that  of  the  stream  (Art. 
154,  Cor.),  but  the  efficiency  is  low,  never  exceeding  0.5. 

Wheels  of  this  kind  are  cumbrous.  In  the  early  days  of 
hydraulic  machines,  they  were  often  used  for  the  sake  of 
simplicity.  In  mountain  countries,  where  unlimited  power 
is  available,  they  are  still  found.  The  water  is  then  con- 
ducted by  an  artificial  channel  to  the  wheel,  which  some- 
times revolves  in  a  horizontal  plane.  When  of  smaU 
diameter,  their  efficiency  is  still  further  diminished.* 

167.  Work  of  the  Poncelet  Water  Wheel.— When 

the  fall  does  not  exceed  G  feet,  the  best  water  motor  to  adopt 
in  many  cases  is  the  Poncelet  undershot  water  wheel.  In 
the  common  undershot  water  wheel,  the  paddles  are  flat, 
whereas  in  the  Poncelet  wheel  they  are  curved,  so  that  the 
direction  of  the  curve  at  the  lower  edge,  where  the  water 
first  meets  the  paddle,  is  the  same  as  the  direction  of  the 
stream.  By  this  arrangement,  the  water,  which  is  allowed 
to  flow  to  the  wheel  with  a  velocity  nearly  equal  to  the 
velocity  due  to  the  whole  fall,  glides  up  the  curved  floats 
without  meeting  with  any  sudden  obstruction,  comes  to 
relative  rest,  then  descends  along  the  float,  and  acquires  at 
the  point  of  discharge  from  the  float  a  backward  velocity 
relative  to  the  wheel  nearly  equal  to  the  forward  velocity  of 

•  See  Cotterill's  App.  Mechs. ;  also,  Fairbaim's  Millwork  and  Machinery. 


WORK  OF  THE  PONCELET   WATER    WHEEL.         287 

the  wheel.  The  water  will  therefore  drop  off  the  floats  de- 
prived of  nearly  all  its  kinetic  energy.  Nearly  the  whole  of 
the  work  of  the  stream  must  therefore  have  been  expended 
in  driving  the  float ;  and  the  water  will  have  been  received 
without  shock,  and  discharged  without  velocity. 

Let  V  and  V  be  the  velocities  of  the  stream  and  float  re- 
spectively ;  then  the  initial  velocity  of  the  stream  relative 
to  the  float  is  v  —  V,  and  the  water  will  continue  to  run  up 
the  curved  float  until  it  comes  to  relative  rest ;  it  will  then 
descend  along  the  float,  acquiring  in  its  descent,  under  the 
influence  of  gravity,  the  same  relative  velocity  which  it  had 
at  the  beginning  of  its  ascent,  but  in  a  contrary  direction. 
Therefore  the  absolute  velocity  of  the  water  leaving  the 
float  is  V—{v  —  V)  =  2  F  —  v. 

Now  the  useful  work  U  done  on  the  wheel  must  equal 
the  work  stored  in  the  water  at  first,  diminished  by  the 
work  stored  in  the  water  on  leaving  the  wheel;  hence 

W  W 

2g  2g^ 

=  iZ  (y  _  F)  F.  (1) 

9 

Comparing  this  expression  with  (1)  of  Art.  156,  we  see 
that  the  work  performed  by  the  Poncelet  wheel  is  double 
that  of  the  common  undershot  wheel. 

ScH. — This  wheel  works  to  the  best  advantage  when  the 
speed  of  the  periphery  is  one-half  that  of  the  stream  (Art. 
154,  Cor.).  This  conclusion  also  follows  from  the  form  of 
the  floats,  as  above  described  ;  since  if  all  the  work  is  taken 
out  of  the  water  when  it  leaves  the  floats,  its  velocity  must 
then  be  zero,  and  therefore  2  F—  y  =:  0,  or  V  =i  |^t'.* 

The  efficiency  of  a  Poncelet  wheel  has  been  found  in  ex- 


♦  The  inventor,  Poncelet,  states  that,  in  practice,  the  velocity  of  the  water,  in 
order  to  produce  its  maximum  eflTect,  ought  to  he  about  2^  times  that  of  the  wheel, 
and  that  the  efficiency  of  the  wheel  is  about  0.7  (Tate's  Mech.  Phil.,  p.  813). 


288  THE  REACTION   WHEEL;   BARKER'S  MILL. 

periraeiits  to  reach  0.68.  It  is  better  to  take  it  at  0.6  in 
estimating  the  power  of  the  wheel,  so  as  to  allow  some 
margin. 

158.    The    Reaction    Wheel;    Barker's    Mill.— 

Pig.  85  shows  a  simple  reaction  wheel.     ACB  is  a  tube? 
capable  of  revolving  about  its  axis,  which 
is  vertical,  and  having  a  horizontal  tube  '  k\\    ' 

DBE  connected  with  it.     Water  is  sup-  €:&^ 

plied  at  C,  which  descends  through  the  O^ 

vertical  tube,  and  issues  through  the  ori- 
fices D  and  E  at  the  extremities  of  the 
horizontal  tube,  so  placed  that  the  direc- 
tion of  motion  of  the  water  is  tangential 
to  the  circle  described  by  the  orifices.  ^ig.^ 

The  efflux  is  in  opposite  directions  from 
the  two  orifices ;  as  the  water  flows  through  BD,  the  press- 
ures on  the  sides  balance  each  other  except  at  D,  where 
there  is  an  uncompensated  pressure  on  the  side  opposite  the 
orifice ;  the  effect  of  this  pressure  or  reaction  is  to  cause 
motion  in  a  direction  opposite  to  that  of  the  jet.  The  same 
effect  is  produced  by  the  water  issuing  at  E,  and  a  continued 
rotation  of  the  machine  is  thus  produced  by  the  reaction  of 
the  jet  in  each  arm. 

Let  h  be  the  available  fall,  measured  from  the  level  of  the 
water  in  the  vertical  pipe  to  the  centres  of  the  orifices,  v 
the  velocity  of  discharge  through  the  jets,  and  Fthe  veloc- 
ity of  the  orifices  in  their  circular  path.  When  the  machine 
is  at  rest,  the  water  issues  from  the  orifices  with  the  velocity 
V^gh  (neglecting  friction).  But  when  the  machine  ro- 
tates, we  have  for  the  velocity  of  discharge  through  the 
orifices,  from  (1)  of  Art.  89, 

V  =  V  F2  +  2gh.  (1) 

While  the  water  passes  through  the  orifices  with  the  ve- 
locity V,  the  orifices  themselves  are  moving  in  the  opposite 


THE  REACTION   WHEEL;   BARKER'S  MILL.  289 

direction  with  the  velocity  V.    The  absolute  velocity  of  the 
water  is  therefore 


V—  V=  VV^+2gh—  V.  (2) 

Now  the  useful  work  done  per  second  by  each  pound  of 
water  must  equal  the  work  due  to  the  height  h,  diminished 
by  the  work  remaining  in  the  water  after  leaving  the 
machine.    Hence, 

useful  work  =  h  —  ^^ — - — — 

^(V^TfcZ)Z,£,om(2),  (3) 

9  ^  * 

The  whole  work  expended  by  the  water  fall  is  h  foot- 
pounds per  second  ;  consequently,  to  find  the  efficiency  of 
the  machine,  we  divide  (3)  by  h  (Anal.  Mechs.,  Art.  216), 
and  get 

efficiency  =  (^il+^^ili:  (5) 

=  1  -  1^,  +  etc.  (6) 

(by  the  Binomial  Theorem), 

which  increases  towards  the  limit  1  as  F  increases  towards 
infinity.  Neglecting  friction,  therefore,  the  maximum 
efficiency  is  reached  when  the  wheel  has  an  infinitely  great 
velocity  of  rotation.  But  this  condition  is  impracticable  to 
realize  ;  and  even  at  practicable  but  high  velocities  of  rota- 
tion, the  prejudicial  resistances,  arising  from  the  friction  of 
the  water  and  the  friction  upon  the  axis,  would  considera- 
bly reduce  the  efficiency.  Experiment  seems  to  show  that 
the  best  efficiency  of  these  machines  is  reached  when  the 
velocity  is  that  due  to  the  head,  so  that  F^  =  'igh. 


290  THE   CENTRIFUGAL   PUMP. 

When  F^  =  "Zgh,  we  have,  from  (5),  neglecting  friction, 
efficiency  =  ^^  '        =  0.828,  (7) 

about  17  per  cent,  of  the  energy  of  the  fall  being  carried 
away  by  the  water  discliarged.  The  actual  efficiency  real- 
ized of  these  machines  appears  to  be  about  60  per  cent,  so 
that  about  23  per  cent,  of  the  whole  head  is  spent  in  over- 
coming frictional  resistances,  in  addition  to  the  energy 
carried  away  by  the  water. 

ScH. — The  reaction  wheel  in  its  crudest  form  is  a  very 
old  machine  known  as  "  Barker's  Mill.''  It  has  been  em- 
ployed to  some  extent  in  practice  as  an  hydraulic  motor,  the 
water  being  admitted  below  and  the  arms  curved.  In  this 
case  the  water  is  transmitted  by  a  pipe  which  descends  be- 
neath the  wheel  and  then  turns  vertically  upwards.  The 
vertical  axle  is  hollow,  and  fits  on  to  the  extremity  of  the 
supply  pipe  with  a  stuffing  box.  In  this  construction  the 
upward  pressure  of  the  water  may  be  made  equal  to  the 
weight  of  the  wheel,  so  that  the  pressure  upon  the  axis  may 
be  nothing.  These  modifications  do  not  in  any  way  afEect 
the  principle  of  the  machine,  but  the  frictional  resistances 
may  probably  be  diminished. 

159.  The  Centrifugal  Pump.— When  large  quanti- 
ties of  water  are  to  be  raised  on  a  low  lift,  no  pump  is  so 
suitable  as  a  centrifugal  pump.  In  this  pump,  water  is 
raised  by  means  of  the  centrifugal  force  given  to  the  water 
in  a  curved  vane  or  arm,  proceeding  from  the  vertical  axis. 
The  dynamic  principles  of  this  machine  are  the  same  as 
those  of  the  reaction  wheel  (Art.  158)  ;  but  they  differ  in 
their  objects.  In  the  latter  machine,  a  fall  of  water  gives  a 
rotatory  motion  to  a  vertical  axis,  while  in  the  former  a 
rotatory  motion  is  given  to  a  vertical  axis  in  order  to  ele- 
vate a  column  of  water. 


THE   CENTRIFUGAL  PUMP.  391 

Let  h  be  the  height  to  which  the  water  is  raised,  meas- 
ured from  the  level  of  the  water  in  the  well  to  the  centre 
of  the  orifice  of  discharge,  v  the  velocity  of  discharge  through 
the  orifice,  and  V  the  velocity  of  the  orifice  in  its  circular 
path,  as  in  Art.  158.  Then  the  work  due  to  the  centrifugal 
force  must  equal  the  work  of  raising  the  water  through  the 
height  h,  increased  by  the  work  stored  in  the  water  at 
efflux ;  therefore 

.-.     V  =  VV^  —  2gh,  (1) 

and  v  —  V=  VV^  —  2gh  —  V 

[as  in  (2)  of  Art.  158]. 

Now  the  work  applied  per  second  to  raise  each  lb.  of 
water  must  equal  the  work  in  raising  the  water  through  the 
height  Ji,  increased  by  the  work  remaining  in  the  water 
after  leaving  the  machine.     Hence 

(i)  _  YY 
applied  work  =  h  -{-  - — - — - 
2g 

9  '  \  f 

The  useful  work  is  h  foot-pounds  per  second ;  therefore 

efficiency  =  7 — —        —  (3) 

{V-'\/V^-2gh)V  ' 

=  ^-W-2-^^-'  (4) 

which  increases  towards  the  limit  1  as  F  increases  towards 
infinity.  Neglecting  friction,  therefore,  the  maximum 
efficiency  is  reached  when  the  pump  has  an  infinitely  great 
velocity  of  rotation,  as  in  the  case  of  the  reaction  wheel. 


292  TtJRSlNES. 

Cor. — When  V^  =  2gh,  we  liave,  from  (3), 

efficiency  =  0.5. 
When  V^  =  ^gJi,  we  have,  from  (3), 

efficiency  =  —. —.  =  .85. 

2(2  —  'V2) 

When  V^  =  6gh,  we  have,  from  (3), 

efficiency  :=  0.9. 

Hence,  theoretically,  the  centrifugal  pump  has  a  con- 
siderable efficiency  when  the  velocity  of  rotation 
exceeds  the  velocity  due  to  twice  the  height  of  the  col- 
umn of  water  raised. 

ScH. — Centrifugal  pumps  work  to  the  best  advantage 
only  at  the  particular  lift  for  which  they  are  designed. 
When  employed  for  variable  lifts,  as  is  constantly  the'  case 
in  practice,  their  efficiency  is  much  reduced,  and  does  not 
exceed  .5,  and  is  often  much  less. 

The  earliest  idea  of  a  centrifugal  pump  was  to  employ  an 
inverted  Barker's  Mill,  consisting  of  a  central  pipe  dipping 
into  water,  connected  with  rotating  arms  placed  at  the  level 
at  which  water  is  to  be  delivered.  The  first  pump  of  this 
kind  which  attracted  notice  was  one  exhibited  by  Mr.  Ap- 
pold  in  1851,  and  the  special  features  of  this  pump  have 
been  retained  in  the  best  pumps  since  constructed.  The 
experiments  conducted  at  the  Great  Exhibition  on  Appold's 
Centrifugal  Pump  with  curved  arms,  gave  the  maximum 
efficiency  0.68.  But  when  the  arms  were  straight  and  ra- 
dial, tlie  efficiency  was  as  low  as  .24,  showing  the  great 
advantage  of  having  the  curved  form  of  the  arms,  which 
causes  the  water  to  be  projected  in  a  tangential  direction. 

160.  Turbines. — A  reaction  wheel  is  defective  in  prin- 
ciple, because  the  water  after  delivery  has  a  rotatory  veloc- 
ity, in  consequence  of  which  a  large  part  of  the  head  is 


TURBINES.  293 

wasted  (Art.  158).  To  avoid  this,  it  is  necessary  to  employ 
a  machine  in  which  some  rotatory  velocity  is  given  to  the 
water  before  entrance,  in  order  that  it  may  be  possible  to 
discharge  it  with  no  velocity  except  that  which  is  absolutely 
required  to  pass  it  through  the  machine.  Such  a  machine 
is  called  a  Turbine,  and  it  is  described  as  ''outward  flow," 
"inward  flow,"  or ''parallel  flow,"  according  as  the  water 
during  its  passage  through  the  machine  diverges  from,  con- 
verges to,  or  moves  parallel  to  the  axis  of  rotation.* 

Turbines  are  wheels,  generally  of  small  size  compared 
with  water  wheels,  driven  chiefly  by  the  impulse  of  the 
water.  The  water  is  allowed,  before  entering  tbe  moving 
part  of  the  turbine,  to  acquire  a  considerable  velocity ;  dur- 
ing its  action  on  the  turbine  this  velocity  is  diminished, 
and  the  impulse  due  to  the  change  of  momentum  drives  the 
turbine. 

Eoughly  speaking,  the  fluid  acts  in  a  water-pressure 
engine  directly  by  its  pressure ;  in  a  water  wheel  chiefly  by 
its  weight  causing  a  pressure,  but  in  part  by  its  kinetic 
energy,  and  in  a  turbine  chiefly  by  its  kinetic  energy,  which 
again  causes  a  pressure,  f 

In  the  outward  and  inward  flow  turbines,  the  water  en- 
ters and  leaves  the  turbine  in  directions  normal  to  the  axis 
of  rotation,  and  the  paths  of  the  molecules  lie  exactly  or 
nearly  in  planes  normal  to  the  axis  of  rotation.  In  outward- 
flow  turbines  the  general  direction  of  flow  is  away  from 
the  axis,  and  in  inward-flow  turbines  towards  the  axis.  In 
parallel-flow  turbines,  the  water  enters  and  leaves  the  tur- 
bine in  a  direction  parallel  to  the  axis  of  rotation,  and  the 
paths  of  the  molecules  lie  on  cylindrical  surfaces  concentric 
with  that  axis. 

There  are  many  forms  of  outward-flow  turbines,  of  which 
the  best  known  was  invented  by  J^'ourneyron,  and  is  com- 
monly known  by  his  name.  The  inward-flow  was  invented 
by  Prof.  Jas.  Thomson. 

•  Cotterill'e  App.  Mechs.,  p.  506.  t  Ency.  Brit,,  Vol.  XII.,  p.  530. 


394  EXAMPLES. 

The  theory  of  turbines  is  too  intricate  a  subject  to  be 
considered  in  this  treatise.  For  a  general  classification  of 
turbines,  with  descriptions,  illustrations,  and  discussions  of 
these  machines,  as  well  as  for  a  further  development  of 
hydraulic  machines  in  detail,  the  student  is  referred,  among 
other  treatises,  to  the  following  :  Fairbairn's  Millwork  and 
Machinery,  Colyer's  Water-Pressure  Machinery,  Barrow's 
Hydraulic  Manual,  Glynn's  Power  of  Water,  Prof.  Unwin's 
Hydraulics. 

EXAM  PLE  S. 

1.  In  a  hydrostatic  bellows  (Fig.  70),  the  tube  A  is  -J  of 
an  inch  in  diameter,  and  the  area  DE  is  a  circle,  the  diam- 
eter of  which  is  a  yard.  Find  the  weight  which  can  be 
supported  by  a  pressure  of  1  lb.  on  the  Avater  in  A. 

Ans.  82,944  lbs. 

2.  Describe  the  siphon  and  its  action.  What  would  be 
the  effect  of  making  a  small  aperture  at  the  highest  point 
of  a  siphon  ? 

3.  A  prismatic  bell  is  lowered  until  the  surface  of  the 
water  within  is  6G  feet  below  the  outer  surface;  state 
approximately  how  much  the  air  is  compressed. 

Ans.  To  \  of  its  original  volume. 

4.  If  a  prismatic  bell  10  feet  high  be  sunk  in  sea  water 
until  the  water  rises  half  way  up  the  bell,  find  how  far  the 
top  of  the  bell  must  sink  below  the  surface,  the  tempera- 
ture remaining  the  same. 

Assiuue  the  water  barometer  —  33  feet  for  sea  water. 

Ans.  28  feet. 

5.  In  the  position  of  the  bell  in  Ex.  4,  find  how  much 
air  must  be  forced  into  it  in  order  to  keep  the  water  down 
to  a  level  of  2  feet  from  its  bottom. 

Ans.  0.72  W,  where  W  is  the  weight  of  the  air  in  the 
bell  when  at  the  surface. 


EXAMPLES.  295 

6.  If  a  small  hole  be  made  in  the  top  of  a  diving  bell, 
will  the  water  flow  in  or  the  air  flow  out  ? 

7.  If  a  cylindrical  diving  bell,  height  5  feet,  be  let  down 
till  the  depth  of  its  top  is  55  feet,  find  (1)  the  space 
occupied  by  the  air,  and  (2)  the  volume  of  air  that  must  be 
forced  in  to  expel  the  water  completely,  the  water  barometer 
standing  at  33  feet. 

Ans.   (1)  1.8  nearly;  (2)  ffths  of  the  volume  of  the  bell. 

8.  The  weight  of  a  diving  bell  is  1120  lbs.,  and  the 
weight  of  the  water  it  would  contain  is  672  lbs.  Find  the 
tension  of  the  rope  when  the  level  of  the  water  inside  the 
bell  is  17  feet  below  the  surface  (A  =  33  feet). 

Ans.  676.48  lbs. 

9.  A  cylindrical  diving  bell  of  height  a  is  sunk  in  water 
till  it  becomes  half  full.     Show  that  the  depth  from  the 

surface  of  the  water  to  the  top  of  the  bell  is  A  —  -• 

10.  A  cylindrical  diving  bell,  of  which  the  height  inside 
is  8  ft.,  is  sunk  till  its  top  is  70  feet  below  the  surface  of  the 
water.  Find  the  depth  of  the  air  space  inside  the  bell 
{h  =  33  .feet).  Ans.  2^  feet. 

11.  (1)  Describe  the  action  of  a  common  pump ;  (2) 
distinguish  between  a  lifting  pump  and  a  forcing  pump ; 
'(3)  to  what  height  could  mercury  be  raised  by  a  pump? 

12.  The  length  of  the  lower  pipe  of  a  common  pump 
above  the  surface  of  the  water  is  10  feet,  and  the  area  of 
the  upper  pipe  is  4  times  that  of  the  lower  ;  prove  that  if 
at  the  end  of  the  first  stroke  the  water  just  rises  into  the 
upper  pipe,  the  length  of  the  stroke  must  be  3  feet  7  inches 
very  nearly  {h  =  33  feet). 

13.  If  the  diameter  of  the  piston  be  3  inches,  and  if  the 
height  of  the  water  in  the  pump  be  20  feet  above  the  well, 
what  is  the  pressure  on  the  piston  ?  Ans.  61.2  lbs. 


296  EXAMPLES. 

14.  If  the  diameter  be  ^\  inches,  the  height  of  the  water 
in  the  pump  27  feet  5  inches,  the  lever  handle  4  feet,  and 
the  distance  from  the  fulcrum  to  the  end  of  the  piston  rod 
4  inches,  find  the  force  necessary  to  work  the  pump-handle. 

Ans.  ^\  lbs. 

15.  The  height  of  the  column  of  water  is  60  feet  above 
the  well,  the  piston  has  a  diameter  of  3  inches,  the  pump- 
handle  is  3|^  feet  from  the  fulcrum,  and  the  distance  of  the 
fulcrum  from  the  piston  rod  is  3^  inches  ;  find  the  force 
necessary  to  work  the  pump.  Ans.  15.3  lbs. 

16.  If  the  height  of  the  cistern  above  the  well  be  25  feet, 
the  diameter  of  the  piston  2  inches,  and  the  leverage  of  the 
handle  12 : 1,  find  the  force  necessary  to  use  in  pumping. 

Ans.  2.83  lbs. 

17.  If  the  height  of  the  cistern  be  42  feet,  the  diameter 
of  the  piston  4^  inches,  the  length  of  the  handle  49  inches, 
and  the  distance  of  the  fulcrum  from  the  piston  rod  3^ 
inches,  find  the  force.  Ans.  20.65  lbs. 

18.  The  diameter  of  the  piston  of  a  lifting  pump  is  1  foot, 
the  piston  range  is  2\  feet,  and  it  makes  8  strokes  per 
minute;  find  the  weight  of  water  discharged  per  minute, 
supposing  that  the  highest  level  of  the  piston  range  is  less 
than  33  feet  above  the  surface  in  the  reservoir  (A  =  33 
feet).  Ans.  312.5t  lbs.,  or  about  983  lbs. 

19.  If  in  working  the  pump  of  Ex.  18,  the  lower  level  of 
the  piston  range  be  314-  feet  above  the  surface  in  the  reser- 
voir, find  the  weight  of  water  discharged  per  minute. 

Ans.  187.5ff  lbs. 

20.  In  a  Bramah's  press  FO  =  1  inch,  FH  =  4  inches, 
the  diameter  of  ^1  =  4  inches,  and  diameter  of  C  =  ^  an 
inch  ;  find  the  force  on  A  produced  by  a  force  of  2  lbs, 
applied  at  ff,  Ans.  512  lbs. 


EXAMPLES.  297 

21.  In  one  of  the  Bramah  presses  used  in  raising  the 
Britannia  tube  over  the  Menai  Straits,  the  diameter  of  the 
piston  C  was  1  inch,  that  of  A  20  inches ;  the  force  applied 
to  C  at  each  stroke  was  2\  tons ;  find  the  lifting  force  pro- 
duced by  the  upward  motion  of  A.  Ans.  1000  tons. 

22.  If  the  receiver  be  4  times  as  large  as  the  barrel  of  an 
air-pump,  find  after  how  many  strokes  the  density  of  the 
air  is  diminished  one-half. 

Ans.  Early  in  the  4th  stroke. 

23.  After  a  very  great  number  of  strokes  of  the  piston  of 
an  air-pump  the  mercury  stands  at  30  inches  in  the 
barometer-gauge,  the  capacity  of  the  barrel  being  one-third 
that  of  the  receiver,  prove  that  after  3  strokes  the  height  of 
the  mercury  is  very  nearly  12f  inches. 

24.  A  fine  tube  of  glass,  closed  at  the  upper  end,  is 
inverted,  and  its  open  end  is  immersed  in  a  basin  of  mer- 
cury, within  the  receiver  of  a  condenser;  the  length  of  the 
tube  is  15  inches,  and  it  is  observed  that  after  3  descents  of 
the  piston  the  mercury  has  risen  5  inches;  how  far  will  it 
have  risen  after  4  descents  ? 

Ans.  The  ascent  x  is  given  by  the  equation  — ;-  +  t 

20 
=  I  4-  oT-     If  A  =  30,  a;  =6.1  nearly. 

25.  If  ^  =  35  (Art.  151),  find  the  elastic  force  of  the 
air  in  the  receiver  after  the  5th,  10th,  15th,  and  20th 
strokes,  the  height  of  the  barometer  being  30  inches. 

Ans.  7.119  ins.;  1.689  ins.;  0.401  ins.;  0.095  ins. 

26.  In  the  same  pump,  the  barometer  standing  at  30, 
find  the  number  of  strokes,  (1)  when  the  mercury  in  the 
gauge  rises  to  25  inches,  and  (2)  when  the  rarefaction  is 
1  -j-  100.  Ans.  (1)  6.2;  (2)  16. 

27.  If  a  hemispherical  diving  bell  be  sunk  in  water  until 
the  surface  of  the  water  inside  the  bell  bisects  its  vertical 


298  EXAMPLES. 

radius,  find  the  depth  of  the  bell,  supposing  the  atmos- 
pheric pressure  to  be  14/<i8  lbs.  to  the  square  inch  {h  =  34). 
Ans.  From  surface  to  surface  73.3  feet. 

28.  There  is  a  pump  lifting  water  29  feet  high,  the 
diameter  of  its  piston  is  1  foot,  the  play  of  the  piston  is  3 
feet,  and  the  pump  makes  10  strokes  per  minute  ;  (1)  how 
many  gallons  of  water  will  be  discharged  per  minute,  and 
(2)  what  is  the  pressure  on  the  piston  ? 

Ans.  (1)  147  gals. ;  (2)  1420  lbs. 

29.  Water  flowing  through  a  trough,  2  feet  wide  and  1 
foot  deep,  with  a  velocity  of  10  feet  per  second  falls  upon 
an  overshot  wheel  50  feet  in  diameter.  Find  (1)  the  part 
of  the  fall  operating  by  impulse ;  (2)  the  maximum  useful 
work  of  the  wheel,  the  eflBciency  being  0.70  ;  and  (3)  the 
number  of  revolutions  the  wheel  makes  per  hour  when 
doing  maximum  work. 

Ans.  (1)  1.55  feet ;  (2)  43076.25  ft.-lbs.  per  sec. ;  (3) 
114.59. 

30.  Water  is  furnished  to  a  breast-wheel  at  the  rate  of 
20  cubic  feet  per  second  with  a  velocity  of  8  feet.  The  fall 
is  20  feet  and  the  efiiciency  0.75.  What  is  the  useful  work 
done  by  the  wheel  when  the  periphery  has  a  velocity  of  3, 
4,  and  5  feet  per  second  respectively  ?  (See  Sch.  1,  Art. 
155). 

Ans.  19185.94;  19215.94;  and  19185.9  ft.-lbs.  per  sec. 
respectively. 

31.  What  is  the  useful  work  done  by  an  undershot 
wheel,  40  feet  in  diameter,  making  120  revolutions  per 
hour,  the  velocity  of  the  water  being  20  feet  per  second  and 
the  area  of  the  vanes  being  1^  square  feet  ? 

Ans.  1910  ft.-lbs.  per  second. 

32.  What  is  the  eflBciency  of  a  reaction  wheel  when  the 
water  having  a  head  of  16  feet,  issues  from  the  orifices  with 
a  velocity  of  45  feet  per  second  ?  Ans.  0.8254. 


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